UNIVERSITY
OF FLORIDA
LIBRARIES
ENGINEERING AND PHYSICS
LI BRARY
Theory of
Mechanical Vibration
JOHN WILEY
& SONS, INC.
NEW YORK
LONDON
Theory of
Mechanical Vibration
KIN N. TONG Professor of Mechanical Engineering
Syracuse University
SECOND PRINTING, JUNE, 1963
Copyright © I960 by John Wiley & Sons, Inc.
All Rights Reserved. This book or any part
thereof must not be reproduced in any form
without the written permission of the publisher.
Library of Congress Catalog Card Number: 606460
Printed in the United States of America
V
t
\\
To the memory of my father
\
Preface
This book is the outgrowth of lecture notes for a course given to
beginning graduate students and qualified seniors. Because of this
origin, it is primarily a textbook, although some utility as a reference
volume is also intended.
A course in mechanical vibrations can be organized in one of two
ways, which may be described as problem centered and theorycentered.
This book is written for a theorycentered course, which develops the
basic principles in a logical order, with engineering applications inserted
as illustrations. No attempt is thus made to cover all problems of
technological importance or to restrict the discussion only to topics
having immediate applications. It is felt that a theorycentered course
has its place in an engineering mechanics curriculum, since the ana
lytical aspects of the theory have pedagogical values beside their
utility in solving vibration problems.
The book is divided into four chapters. Chapter 1 treats systems
having a single degree of freedom. All the basic concepts pertaining
to mechanical vibrations are presented, with the exception of vibration
modes. Chapter 2 introduces the concept of vibration modes in a
multidegreefreedom system, using a system with two degrees of free
dom as a simple model. The discussion is kept as close as possible to
physical aspects of the problem. By means of matrix algebra and
generalized coordinates, Chapter 3 extends the results previously ob
tained. In this way this chapter also lays the foundation for the solu
tion of vibration problems on digital computers and provides a heuris
tic picture of what is to follow. Chapter 4 discusses the vibration of con
tinuous media. Because only a limited amount of student knowledge
in elasticity can be assumed, the systems selected for illustration in this
chapter are relatively simple, yet the theory presented is quite general.
vii
Vlll PREFACE
The layout of this book is somewhat different from the usual. In the
beginning of each chapter fundamental principles are presented in a
connected series of articles. Articles dealing with examples, applica
tions, and specialized topics, which are more or less independent of one
another, are placed at the ends of the chapters. (In the first three
chapters these articles are grouped into two sections, A and B. The
same grouping is not indicated in the fourth chapter, since there the
demarcation is not so clear.) The purpose of this arrangement is two
fold. It emphasizes the structural coherence of the theory, and it
affords flexibility in classroom assignments. The instructor can plan
his lectures by following the main development of the theory. At
intervals appropriate to the level and the interest of a particular class,
he may discuss, or assign as home reading, examples, applications, and
methods selected from this book or from other sources. A number of
exercises is given at the end of each chapter. Many of' these exercises
supplement the material in the text.
The students are assumed to have the usual preparation, including a
course in differential equations, in undergraduate mechanics and mathe
matics. Certain fundamental theorems in advanced calculus and in
vector analysis are referred to in a few isolated passages; these can be
omitted, if necessary, without disrupting the continuity of presenta
tion. An appendix on the basic ideas of matrix algebra is given. The
scope of this appendix is limited, but it contains all that is needed for
studying Chapter 3. In short, little prior knowledge is required to
understand this book, although some degree of maturity is indispens
able.
To keep the scope of the book within the limits of a twosemester
course and to preserve the unity of the entire presentation, certain
topics are omitted. These include nonlinear vibrations and the solution
of transient problems by operational calculus. However, seeding ideas
pertaining to these topics are planted in Arts. 1.4, 1.10, 1.11, 1.13, 1.14,
2.8 and 3.8, but their complete development is left to other standard
courses generally available to advanced students.
Many persons helped to prepare this book. I wish especially to
thank Professor Harold Lurie for a thorough reading of the manuscript
and for offering valuable suggestions. Thanks are due to Mrs. Patricia
Fisch and Mrs. Marilyn Levine for typing the manuscript and to Mr.
C. Y. Chia and Mr. K. Ruei for assisting in various other ways.
Kin N. Tong
Syracuse, New York
October 1959
Contents
Introductory Remarks
CHAPTER 1
of Freedom
Systems with a Single Degree
Section A Theory and Principles
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
Section B
1.12
Introduction 3
Simple harmonic motion 3
Complex number and graphical representation of
a sinusoidal function 6
Harmonic oscillation of system with a single
degree of freedom — General discussion 9
Energy relation, Rayleigh's principle, and
phase trajectory 10
Damped vibration with viscous or linear
damping 12
Forced vibration under a harmonic force 17
Complex number representation 24
Steadystate response to periodic forces 28
Work done by external forces and energy
dissipation in vibratory systems 30
Response of linear systems to a general external
force — Superposition theorem 31
Signalresponse relation of linear systems in
general 39
Methods and Applications
Examples of linear vibratory systems with a
single degree of freedom 52
52
CONTENTS
1.13 Linearization of systems in small oscillations 55
1.14 Piecewiselinear systems 59
1.15 Theory of galvanometer and movingcoil
instruments 68
1.16 Seismic instruments and transducers 73
1.17 Vehicle suspension 79
1.18 Structural damping and the concept of complex
stiffness 82
Exercises 89
CHAPTER 2
of Freedom
Systems with Two Degrees
Section A
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Section B
2.11
2.12
2.13
2.14
Theory and Principles 100
Introduction 100
Free undamped vibration — a model and its
equation of motion 100
Principal or normal modes 101
General Solution 102
Formulation by energy consideration — a
generalized analysis for the free vibration of
system with two degrees of freedom 105
The use of influence coefficients 108
Rayleigh's quotient 111
Vibration of damped systems 114
Forced vibration 116
Degenerated cases 120
Repeated roots in frequency equations —
transverse vibration of rotating shafts 123
Methods and Applications 128
Illustrative examples 128
Application of Rayleigh's method 132
Some principles in vibration control 135
Effects of rotation on critical speeds of shafts 146
Exercises 161
CHAPTER 3 Systems with Several Degrees
of Freedom
Section A
3.0
Theory and Principles
Introduction 168
168
CONTENTS XI
3.1 Generalized coordinates, constraints, and degrees
of freedom 169
3.2 Energy expressions in generalized coordinates for
linear systems 170
3.3 Summation convention and matrix notation 173
3.4 Free vibrations of an undamped system — an
eigenvalue problem 175
3.5 Principal coordinates and orthogonal property of
modal vectors 180
3.6 Rayleigh's quotient 183
3.7 Forced vibration of an undamped system 185
3.8 Free and forced vibrations of a damped
system 188
3.9 Semidefinite systems 192
3.10 Repeated roots of the frequency equation 196
Section B Methods and Applications 197
3.11 Solution of eigenvalue problems by matrix
iteration 197
3.12 Additional theorems and methods 208
3.13 Chain systems — Holzer's method 212
3.14 Electrical analog of mechanical systems and
electromechanical systems 221
Exercises 231
CHAPTER 4 Vibration of Elastic Bodies
4.0 Introduction 236
4.1 Coordinates and constraints 237
4.2 Formulation of a problem by differential
equation 239
4.3 Separation of time variable from space variables —
reduction to eigenvalue problems 249
4.4 Orthogonal property of eigenfunctions 260
4.5 Formulation by integral equation 271
4.6 Rayleigh's quotient and its stationary values 274
4.7 RayleighRitz method 279
4.8 Formulation of problem by infiniteseries expan
sions of energy expressions — RayleighRitz
method reexamined 287
4.9 Forced vibration of elastic bodies 296
4.10 Vibration of an infinite or semiinfinite elastic
body — wave phenomenon 304
xii CONTENTS
4.11 Methods of finite differences 312
Exercises 318
APPENDIX. Outline of Matrix Algebra in
Linear Transformation of Vectors 323
INDEX 343
Introductory Remarks
Vibratory motions in machines and structures are of frequent concern
in engineering practice. Such motion is usually objectionable; it may
sometimes be desirable; and frequently its presence reveals to us the
inner workings of complex machinery in operation. Whether our object
is to minimize vibrations, to enhance them, or to utilize them for "diag
nostic" purposes, it is essential that the physical laws which govern such
motions be studied.
Vibratory motions are essentially periodic in character. Theory of
vibration is therefore a special topic in analytical dynamics dealing
principally with periodic motions of mechanical systems. In recent times,
however, we have come to appreciate that the susceptibility of a mechani
cal system to vibrations describes certain intrinsic properties of the system,
from which its general dynamic behavior can often be deduced even when
periodic motion is not involved. Thus in a subtle way the principles
governing the vibratory behavior of a system may often be advantageously
used to supplement or even to replace the differential equations of motions
in analyzing a number of problems in dynamics.
Theory of vibration also claims kinship in varying degrees of removedness
with such studies as acoustics, alternating current circuits, and electro
magnetic waves. All these studies concern themselves in part with periodi
cally changing phenomena which are governed by more or less unified
principles; and the analytical results obtained in these disciplines often
have interchangeable applicabilities. Furthermore, if we examine some of
the analytical tools used in theory of vibration, we will also find that they
resemble those used in a still larger class of physical and mathematical
problems. These include problems in elastic instability, periodic structures
and linear transformations.
For the various reasons stated it then becomes justified to lift the study
of vibratory motions from its parent body, analytical dynamics, and to
1
2 THEORY OF MECHANICAL VIBRATION
place it on a pedestal of its own. In developing this subject, we have
therefore not confined our attention strictly to the utility of the theory
in solving vibration problems but have cast occasional glances upon many
of its neighboring domains, into which we are likely to excurse in our
future studies. Consequently, certain terms and concepts are introduced,
not because of any immediate bearing they have upon the subject on hand
but because of the desirability of acquainting ourselves with the many
bridges that connect seemingly unrelated fields.
CHAPTER 1
Systems with a Single
Degree of Freedom
SECTION A. THEORY AND PRINCIPLES
1.0 Introduction
Our study begins with the discussion of vibrations of a system having
only a single degree of freedom. We shall, however, refrain from defining
at this moment precisely what is meant by such terms as "vibrations,"
"systems," and "degree of freedom." Instead, we shall try to build up our
theory by analyzing some simple and readily visualized physical systems
and leave the precise definitions of these terms to more opportune moments.
In the meantime, terminologies not specifically defined are to be inter
preted according to their accepted meanings in mechanics, in mathematics,
and in the English language.
1.1 Simple Harmonic Motion
The simplest form of vibratory motion is a simple harmonic motion,
which is defined in kinematics as a rectilinear motion of a point whose
acceleration at any time t is proportional and opposite to its displacement
x. The mathematical description of this motion is therefore the differential
equation
dH
W* = ° JX
or 1 x + w 2 x = (1)
1 The symbols x and x stand for the first and second time derivatives of the function
x(t), respectively. This notation, employed originally by Newton, is used throughout
this book, except when ambiguity may arise.
3
4 THEORY OF MECHANICAL VIBRATION (1.1)
in which co is a real number, hence the constant of proportionality, co 2 ,
is always negative. A simple example of a motion described by (1) is that
of the mass particle in the system shown in Fig. 1. This system consists of a
linear coil spring having a spring constant k and a mass particle of mass
m, which is attached to the spring and constrained to move along the axis
of the spring. Let be the position occupied by m when all the external
forces acting on m are in equilibrium, and let x be the displacement of m from
at time t. If the displacement x causes a change only in the force exerted
by the spring while all the other forces on m remain constant, 2 then according
W&
S
I
m
I
oA
Figure 1
to Hooke's law of elasticity and Newton's law of motion the acceleration
of m due to its departure from its equilibrium position is given by
mx = — kx
or
mx + kx = (2)
If we let
«>*= (3)
m
we reduce (2) into the form of (1).
The solution of the differential equation (1), that is, the relationship of x
versus t which obeys (1), is known to be expressible in the general form
x(t) = A cos cot + B sin cot (4)
2 According to the way we formulate this problem, the question whether or not there
is a gravity force is irrelevant.
(1.1) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 5
This expression has two constants of integration, A and B, which are not
contained in the original differential equation and are therefore not
determined by it. In other words, the physical constants of the system
alone do not specify the motion uniquely without some other information
concerning the actual motion itself. This information may be furnished
by the initial condition, 3 that is, the displacement x and the velocity x
at the instant t = 0. Substituting this condition in (4), and in the equation
obtained by differentiating (4), we have
x = A x = Bco (5)
It is sometimes more convenient to express the solution in the form
x{t) = C cos (cot — a) (6)
where
C cos a = A C sin a = B
or
C 2 = A 2 + B 2 a = tan 1  (7)
A
In a simple harmonic motion the displacement is therefore a sinusoidal
function of time, and the motion is a periodic oscillation characterized
by the quantities co, C, and a. The quantity co, which is determined by the
properties of the system, is called the circular frequency. It is related to the
frequency f and the period T by
co = 2nf T= = — (8) 4
/ 0)
The quantity C, which is the maximum displacement in the oscillatory
motion, is called the amplitude. Ordinarily, frequency and amplitude taken
together are enough to give a complete description of a harmonic oscilla
tion, just as pitch and intensity are enough to describe a pure musical tone.
The need here for a third quantity a, which is called the timephase angle,
therefore deserves further explanation. We see from (6) that a is an angle
whose cosine gives the ratio of the displacement at t = to the amplitude
of vibration. Hence the value of a depends on the designation of an
instant corresponding to t = 0. For a truly periodic phenomenon, which,
3 Although the term "initial condition" has the connotation of the condition at the
beginning of the motion, the condition at any specific instant can be used for this
purpose, and no generality is lost in setting / = for the instant in question.
4 The difference between to and /is merely a matter of units. The conversion factor
is the constant In radians per cycle. Hence, whenever this difference is immaterial tq
our discussion, we use the term frequency for co also.
6 THEORY OF MECHANICAL VIBRATIONS (1.2)
as described by (1) and (6), has no beginning and no end, this designation
is an arbitrary act, and the value of a has very little physical significance.
If, however, the relations (1) and (6) come into effect only at the instant
/ = 0, when a sudden change of external conditions affecting the system
takes place, then the timephase angle a contributes toward the description
of this initial condition. For instance, if the springmass system is set into
motion at t = by external means acting momentarily to give the mass a
displacement x Q and velocity x , these quantities are then related to y.
through the equation
B x a
tan a =  = —  (9)
A x co
When two periodic phenomena of the same frequency are studied
together, the difference between their timephase angles is called the phase
difference. The phase difference measures how far these two periodic
phenomena are out of step with each other. In many problems this phase
difference rather than the absolute value of the timephase angle has
physical meaning.
1.2 Complex Number and Graphical Representation of a Sinusoidal
Function
It is known in algebra that trigonometric functions are related to
exponential function by Euler's formula:
e i0 = cos + i sin
For real values of 6 the real part of the complex number e* fl is cos 8. If
we use the symbol "Re" to mean "the real part of." we may write (6) as
x = C Re (V (w ' a) ) = Re fCe^'V" 1 )
(10)
= Re (Ae lMt )
where
/ = Ce~ H = C cos a — iC sin a = A — iB (11)
This quantity / is called the complex amplitude of the sinusoidal function
<t).
At this point it seems unnecessarily complicated and highly artificial to
represent a cosine function by the real part of a complex exponential
function. Certain advantages of this representation will, however, reveal
themselves in time.
A complex number
a + ib = re id
(1.2)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
can be represented by a vector in the socalled Argand's diagram, as
shown in Fig. 2. The length of the vector r is called the absolute value (or
modulus) and the angle 6, the argument (or amplitude)^ of the complex
number in question. Hence the absolute value of A is the amplitude of
x(t), and the argument of A is the timephase angle of x(t). In this way A
gives a complete description of the integration constants, whereas the
parameter of the original differential equation, viz., a>, is contained in the
factor e iuit . In (10) we may say that e lwt represents the differential equation
(or the system) and A represents a particular solution of the equation (or a
particular motion of the system).
/
Imaginary
(a + ib)
>^
b
^V
a
Figure 2
In Argand's diagram representation the complex amplitude A is shown
as OL in Fig. 3. The complex number whose real part represents x at
time / is represented by rotating OL through an angle cot to OX. The value
of a: is then equal to the projection of OX on the real axis as it rotates around
O with uniform angular velocity co. The vector OX is called a rotating
vector. 6 When two or more rotating vectors with the same to are involved
in a problem, it is only their relative positions that are usually of import
ance. Hence their positions may be "frozen" at a chosen instant, say
/ = 0, when their relation is studied. For instance, if x(t) is a sinusoidal
function having the frequency w and is represented by a rotating vector
Xe iwt , then x and x are represented by rotating vectors 90 and 180° "ahead"
of Ae ltot , respectively. The relative positions of these rotating vectors do
5 It is unfortunate that the word "amplitude" is used both in the description of a
complex number and in that of an oscillatory phenomenon. It becomes doubly unfor
tunate that the amplitude of one does not correspond to that of the other when a
complex function is used to describe an oscillatory phenomenon. For this reason we
use the word "amplitude" only in describing an oscillation and the word "argument"
in describing a complex number.
Those who are interested in the vagary of meaning of the word "amplitude" should
also look up its definition in astronomy.
6 It is also called a phasor or a sinor.
8
THEORY OF MECHANICAL VIBRATIONS
(1.2)
not change as they rotate en masse with the same angular velocity ca.
The position of a rotating vector at / = is, of course, the vector represent
ing the complex amplitude; therefore we often use the complex amplitude
alone to represent a sinusoidal function of time if the frequency m is known
to be constant and is of little consequence in the discussion.
Figure 3
It can easily be verified that the sum of two complex numbers is repre
sented in Argand's diagram by a vector that is the vectorial sum of the two
vectors representing the two numbers. Furthermore, a little analysis (see
Exercise 1.5) will show that superposition of two sinusoidal functions of
time having the same frequency yields another sinusoidal function of the
same frequency, whose complex amplitude is the sum of those for the
original functions. Hence parallelogram theorem in Argand's diagram
represents the rule for the superposition of sinuosidal functions of the
same frequency. Rotating vectors of the same circular frequency therefore
(1.3) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 9
have the properties of vectors insofar as addition law is concerned.
However, they do not possess all of the properties of vectors in the usual
sense.
1.3 Harmonic Oscillation of Systems with a Single Degree of Freedom —
General Discussion
The discussion so far is essentially a study of the solution (and of its
representations) of a differential equation. The springmass system shown
in Fig. 1 is merely an illustration of a physical system whose motion
is described by the differential equation. Needless to say, the results
obtained in this study, being mathematical in nature, are applicable to the
motions of all other systems in which the same mathematical formulation
is valid. 7 We now discuss this class of systems in general.
The springmass system illustrated is said to have a single degree of
freedom because only a single variable is needed to specify the configura
tion of the system. By configuration we mean the location of all mass
particles of the system in space. 8 A system generally consists of many,
or infinitely many, mass particles. But in a system with a single degree of
freedom the spatial coordinates of these mass particles are interrelated to
one another and restricted to vary in such a way that only a single (but not
unique) quantity is required to specify their locations. This restriction in
their movements is supplied by what we call the constraints of the system.
In the springmass system studied we have tacitly assumed that (a) the
mass m is a rigid body, (b) external things, such as guides, restrict the
motion of m to a translation, (c) the spring is weightless. 9 Under these
assumptions, the configuration of this system at any instant can be repre
sented by a single function of time, x(t), called the coordinate of the system,
for which we chose the displacement of m from its equilibrium position.
Other choices can be made to serve our purpose equally well ; but, what
ever the particular choice may be, physical reasoning leads to the con
clusion that it must be a periodic function of time having the identical
7 This observation may sound trite but upon it rests the most imposing edifice of
mathematical physics.
8 This means that in this definition we are concerned only with the geometrical state
of the system, not with its general dynamic state, which involves also the velocities or the
momentum of the mass particles, discussed in Art. 1.4.
9 If you do not like the idea of weightless things, we can compromise the situation
in the following manner. We assume that the mass of the spring is relatively small so
that its deformation is largely produced by the forces acting at its two ends without any
measurable contribution from the inertia force of the spring itself. In that case the
displacement of each particle of the spring is determined by its total elongation and the
laws of elastic deformation. (See also Art. 4.0.)
10 THEORY OF MECHANICAL VIBRATIONS (1.4)
frequency. In other words, the frequency of a vibratory system is a physical
property independent of the choice of the coordinate.
A number of vibratory systems with a single degree of freedom are
illustrated in Arts. 1.12 and 1.13. However, for our present exposition we
may continue to use the springmass system as a model without the loss of
generality.
1.4 Energy Relation, Rayleigh's Principle, and Phase Trajectory
The systems being studied are called conservative systems because they
contain no mechanism for the dissipation of mechanical energy. Therefore,
once such a system is set into vibration, it contains a fixed amount of
mechanical energy. This amount is divided between the potential and the
kinetic energy. The vibration of the system can be considered as a periodic
transferring back and forth of the energy from one form to the other.
At the instant the system assumes its equilibrium configuration, its
potential energy is a minimum, 10 or zero, and its kinetic energy, a maximum.
When the system is momentarily at rest, the reverse is true. The maximum
potential energy and the maximum kinetic energy occurring at these two
extreme configurations obviously must be equal.
Using T and V as symbols of kinetic energy and potential energy,
respectively, we have for a simple springmass system
T = imx 2 V = \kx 2
Since the motion is a harmonic oscillation and the maximum kinetic
energy T m and the maximum potential energy V m are equal, we have
T m = imco 2 C 2
V m = \kC*
and
T m = V m or to = Vkf^i (12)
In many problems in which to is to be determined it is easier to express the
energies in terms of the amplitude C than to write the equation of motion,
especially when approximations are involved. This method of determining
the natural frequency of a system in harmonic oscillation is called Rayleigh's
method. Its application to vibration problems is illustrated in Art. 1.13
The reader is advised to study these illustrative problems before pro
ceeding further.
10 Recall the theorem in mechanics which states that the necessary condition for a
system in stable equilibrium is that its potential energy is at a minimum.
(1.4) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 11
In a more general discussion both m and k in the energy expressions
must be considered as functions of the coordinates x. Hence 11
T = \m(x)x 2 V = \k(x)x 2
For conservative systems the conservation of energy demands that
T + V = constant
or
\m{x)x 2 + \k(x)x 2 = \h 2
h l \m h 2 \k
in which h is a constant.
Consider a coordinate system in which the coordinate axes are the
displacement x and velocity x. A generic point P in this coordinate plane,
which is called the phase plane of motion, represents a dynamic state of
the system, specified by the displacement and velocity taken together. The
motion of the physical system is represented by the motion of P in the
phase plane. The locus traced by P is called the phase trajectory, and the
velocity of P in the phase plane along the phase trajectory is the phase
velocity. Much about the motion of a system can be revealed by studying
the topological structure of its phase trajectory in the phase plane.
The phase trajectories of a given system are described by (13), in which
m and k are determined by the system and h is a parameter determined
by the initial condition (or energy content) of the system. For conservative
systems in oscillation the phase trajectory must be a closed curve. The
totality of all possible oscillations for a given system is represented by a
oneparametric family of such curves.
According to (13), the phase trajectory of the motion of a springmass
system in which m and k are constants is an ellipse, shown in Fig. 4. The
ratio of the two axes of the ellipse is the circular frequency of the system,
and their absolute lengths are determined by the total energy content of
the system in motion. Therefore, one may say that the constants of the
system determine the shape of the ellipse, and the initial conditions, the
size. 12 For a given system, through each point in the phase plan except
11 How x and x enter into these expressions is discussed more fully in Chapter 3 and
is of no importance to our present discussion. See also Art. 1.13 and Exercise 1.26.
12 It is an important property of a linear system that its natural frequency is independ
ent of its amplitude. The definition of the term "linear system" is discussed more
thoroughly in Art. 1.10.
12
THEORY OF MECHANICAL VIBRATIONS
(1.5)
the origin, there passes a unique phase trajectory (ellipse). By giving the
system an initial condition or dynamic state corresponding to this point,
the resulting motion is depicted by the ellipse passing through this point.
X
1 .

^S
<
v_
_y
„ h
Figure 4
1.5 Damped Vibration with Viscous or Linear Damping
A viscous damper is a device that offers a resistance to motion propor
tional to the time rate of its deformation. If such a device is incorporated
y//////////////^^^^^^
Figure 5
into the springmass system studied, as shown in Fig. 5, the resulting
equation of motion is
mx = —kx — ex
or (14)
mx + ex f kx =
(1.5)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
13
The constant of proportionality c is called the damping constant, or simply
damping. The general solution of (14) is given by
x(t) = Q^ + C 2 e s ^ (15)
where
— c ± Vc 2 — 4mk
2m
(16)
and Q and C 2 are constants of integration to be determined by the initial
condition. There are three possible cases of the resulting motion.
Figure 6
Case 1. Overdamped systems
c 2 — 4mA: >
In this case s x and s 2 are both real and negative. For the motion with the
initial condition
x ^= Xq x ^ = Xq at t ^ = u
the constants are
Q
^2^0 i ^0
C 9 =
SjXq + ^o
(17)
°1 °2 °2 °1
The resulting motion may be one of the types represented in Fig. 6.
Depending upon the relative magnitudes of x and x , the curve may cross
the /axis at most once. (See Exercise 1.20.) At any rate, no oscillation
may take place ; and x and x approach zero asymptotically. The motion is
said to be aperiodic.
Case 2. Critically damped system
c 2 — 4mk = c
= iVmk
14 THEORY OF MECHANICAL VIBRATIONS (1.5)
For this particular value of damping, c c , called the critical damping of the
system, we have
s i = s 2 — ~ 0)
and the solution of (14) becomes
x(t) = C 1 e~ (o1 + C 2 te M (18)
with
Cj = Xq C 2 = x ~r ojXq
The motion does not differ from that of Case 1 in its essential character.
It is also aperiodic.
Case 3. Underdamped system
c 2  4mk <
In this case s x and s 2 are a pair of complex conjugate numbers with
negative real parts. The solution is more conveniently expressed as
x(t) = e {  cl2m)t (A cos co e t + B sin co c t) (19)
or
x(t) = Ce {  Cl2m » cos (o) c t  a) (19a)
where
V4mk — c 2 I ( c 2
1J (20)
2m v \ c v
and the relationships among A, B, C, and a are the same as in (7). The
motion is therefore oscillatory in nature and is often called a damped
oscillation. It can also be conveniently thought of as a sinusoidal motion
with diminishing amplitude. 13
In complex number representation (19) and (\9a) may also be written
x = Re (Xe iat ) (21)
where
X = Ce ix = A  iB
and
a = co c + i — = oi c + i— co (22)
2m c c
The quantities X and cr are called the complex amplitude and the complex
frequency of a damped vibration. From (22) it is seen that a is a property
of the system, whereas ?. describes the initial condition.
13 To say that this description is mathematically unacceptable, since a sinusoidal
motion must have a constant amplitude, is "justified quibbling."
(1.5) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 15
For
X
= x at
/ =
Xq =
Re (A) = A
Xq =
Re {ioX)
iB
. C c X
= x n  I
+
ccox
Hence
X = AiB = x i ~"" u ' u (23)
co c c c
Returning to (19), we see that the ratio of two displacements at any two
instants a period, T = 2rrj 'co c , apart is a constant.
X {*' Jrl9m\T (1A\
x(t + T)
In other words, at regular time intervals of 2ttJ(o c the displacement
decreases in a geometric progression. The natural logarithm of this ratio
is called the logarithmic decrement, denoted by the symbol A.
x(t + T) 2m o) c
The value of A is a measure of the amount of damping the system
possesses. In many actual systems the damping force is not so localized
that the damping constant c can be directly measured. If such a system
is set into vibration, one can observe A and deduce an equivalent damping
constant for the system by the following relationships:
c 2tt c to
A = = 2tt
2m co c c c co c
Electrical engineers, for good reasons, generally prefer to use a quantity
called quality factor Q to describe the damping property of a circuit. The
quantity Q can be defined by
Q = 
* A
From (20) we also have
16
THEORY OF MECHANICAL VIBRATIONS
(1.5)
Thus the three ratios ojJco, c/c c , and A/ln can be expressed as three
trigonometric functions of a single angle d, as shown in Fig. 7.
sin d =
C c
and
c oj c
cos = —
(O
oje
. A
tan o = —
2tt
(27)
Theoretically, the equivalent damping constant can also be obtained
from measuring oj c , but practically this cannot be done with accuracy
when the amount of damping is small. For example, it can be seen in
\^
c
o\
A
2tt
^8
w c
CO
Figure 7
(26) that with a damping of 20 per cent of the critical value there is only
less than 2 per cent difference between eo and co c . This also shows that the
introduction of a moderate amount of damping changes the frequency of a
vibratory system only slightly. Furthermore if the damping is small,
l***± (28)
C c 277
A graphical representation of the motion can now be constructed, as
shown in Figs. 8 and 9. We see from (21) that the displacement x is
represented by the projection on the real axis of the vector in Argand's
diagram representing the complex number Xe [at . This vector rotates with
angular velocity eo c and decreases in length exponentially. The end of the
vector describes a logarithm spiral}* which has the geometric property
that the angle between its radius vector and its normal is constant. Let
it be an exercise (Exercise 1.12) to prove that this angle is the angle 6
defined in (27). (Work Exercise 1.11 also.)
14 A logarithm spiral has many other interesting geometric properties. It fascinated
the famous mathematician Jakob Bernoulli (16541705) so much that he willed it to be
inscribed on his tombstone. Unfortunately, the execution of his will was not done with
mathematical care and the inscription appeared to resemble an Archimedian spiral.
(1.6)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
17
Figure 8
(c/2m)t
Figure 9
1.6 Forced Vibration under a Harmonic Force
(a) undamped system
Let an external harmonic force, that is, a force which is a sinusoidal
function of time, be acting on the springmass system shown in Fig. 1.
The differential equation of motion may be written
mx + kx = F cos co f t
(29)
18 THEORY OF MECHANICAL VIBRATIONS (1.6)
in which F is the amplitude of the force and to f is the circular frequency. 15
The solution of (29) is
f
x = A cos cot + B sin cot + z — z cos a>J (30)
k — moj f 2
in which A and B are constants of integration to be determined by the
initial condition. Let this be x = x and x = x at t = 0. Then
and
^o B = —
k — mco f 2 co
A = x   ^— 2 B =  u (31)
x F F
x = x cos cot + — sin art — —  cos cot +' —  cos co f t (32)
co k — nuof k — mcvf
Several interpretations for the physical meanings of the terms in (32)
are possible. Let us consider the following one. Suppose prior to t =
the system is in free vibration, and at t = 0, when its displacement and
velocity are x and x , respectively, an external force F cos co f t begins to
act on the system. Both the equation of motion (29) and its solution (32)
are then valid for t > 0. Comparing (32) with (4) and (5), one sees that
the first two terms of (32) represent the motion that was in existence at
t = and has the natural frequency of the system co. The third term
represents the motion that is produced by the sudden application of a
force at t = and also has the natural frequency co. These two motions
are therefore due to conditions existing and to things happening at t = 0.
They are called the transients because they eventually die out if there is
any dissipation mechanism in the system, however small. 16 On the other
hand, the motion represented by the last term has the frequency of the
external force and is ever present as long as the force continues to act on
the system. This motion is called the steadystate response of the system
to the force F cos co f t. In many practical problems one is merely interested
in motions that persist. An analysis that ignores the transient terms is
called a steadystate analysis.
15 The physical meanings of the words "amplitude" and "frequency" when applied
to a harmonic force should be obvious.
16 The decay of these transient terms is analogous to that of (19) if there is any damping
force. Physical reasoning also shows that unless a system is completely free from
disturbances or is unstable its condition at time t = 0, which determines the magnitude
of the first three terms in (32), must have decreasingly small influence on its condition
at subsequent instants. Note also that the quantity F in (31) is better interpreted as the
value of the force at t = than the amplitude of the harmonic force.
(1.6) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
The steadystate response of the system being analyzed is thus
»st
k — mco f
2 C0S "V=, _, , tf
1  (m,la>y
cos co f t = A cos 0) f t
19
(33)
where d si = Fjk is called the static deflection, which is the deflection of
the spring under a static force equal to the amplitude of the external
harmonic force. 17
The nondimensional ratio U/<5 S/ , called the magnification factor, 18
versus the frequency ratio co f jco is plotted in Fig. 10. It is seen that for
Figure 10
forcing frequency co f below the natural frequency co, \?.\ and F have the
same sign, and the displacement is in phase with the force. As co f increases
toward co, \k\jb st increases from unity to larger values.
According to (33), for (o f larger than co, F and A have opposite signs.
This is to be interpreted that the displacement is 180° out of phase with the
force. The amplitude decreases with increasing «j f and approaches zero
as a limit. At co f = co the amplitude is indeterminate, and a condition
called resonance exists.
To study what takes place at resonance, let us consider first the case in
which (o f is nearly equal to but slightly smaller than co, or
CO
co f = 2 A co
17 Note carefully the definition of d st . Although it may have other physical meanings
in other special cases, the definition given here is a more general one.
18 Further discussions on "magnification factor" are contained in this article and in
Art. 1.11.
20
THEORY OF MECHANICAL VIBRATIONS
(1.6)
Assuming in (32) that the system is at rest at t = 0, that is, x = i e = 0,
we have
Since
k — mco f 2
(cos CO f t — cos cot)
(oj f + co)t
1  (cof/cof
u 2 d st
Ao) (GO + COf)
COf + CO
2 sin
sin A a> t sin
sin Acd t
(co f + oj)t
> Ao>
(34)
within a few cycles of sin (co f + eo)//2, the function sin Aoj t changes only
slightly. The resulting motion can be represented by a series of sine waves
sin Aut
*t
Figure 11
whose amplitudes change slowly within envelopes which are two sine
waves of frequency 2Aco, 19 shown in Fig. 11.
Returning to (34), we now let Aco— > 0; then
sin Aco t— > Aco /
lim x(i) = b st {cotj2) sin cot
Aco^O
(35)
19 Since sin Aco t enters as a part of the amplitude only, its absolute value alone is
important, and this value repeats itself every 180° instead of 360 \
(1.6)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
21
Thus the amplitude grows indefinitely as / increases. 20 The result is
shown in Fig. 12. This condition is called resonance. It is important to
note that the motion is a sine function, whereas the force is a cosine
function. Hence the displacement lags behind the force by a phase
difference of 90° instead of or 180°, as in the case of nonresonant forced
vibrations.
The motion described by (34) is an example of a phenomenon called
beat. It is formed by two harmonic oscillations with the same amplitudes
*t
Figure 12
but slightly different frequencies. The phase difference between the two
changes continually so that sometimes the two oscillations reinforce each
other and at other times they cancel each other. In a more general beat
phenomenon the amplitudes of the two need not be the same. The general
case is therefore represented by
x(t) = A cos (co + Aco)/ + B cos (co — Aco)/
= (A + B) cos A.co t cos cot — (A — B) sin cot sin Aco t
= C cos (cot — a) (36)
where
C 2 = A 2 + B 2 + 2AB cos (2 Aw /)
tan a =   tan (Aco t)
A + B
(37)
21
20 This is a more precise statement than the one that asserts that the amplitude is
infinite. Aside from physical limitations, such as overstraining and breaking of parts,
it takes time to build up large amplitude and infinite time to build up infinite amplitude.
True steadystate forced vibration of an undamped system is not possible at resonance.
21 In electronics we call the variation of C with t amplitude modulation and that of a
with t, frequency modulation.
22
THEORY OF MECHANICAL VIBRATIONS
0.6)
The foregoing relations are nothing but the law of cosine and the law of
tangent applied to the triangle shown in Fig. 13. The maximum and
minimum "displacements" are therefore approximately the sum and the
difference of the amplitudes A and B, respectively. The beat frequency is
Bisector
of LAOB
2 A art
Figure 13
Aoj/7r, or the difference between the original frequencies. In Fig. 14 the
two different cases of combined motion are shown: (a) co = Aeu and (b)
co ^> Aco. It is to be noted that in (b) the envelopes are not sine waves,
being somewhat "pinched in" in the valleys, especially when A and B are
co = Aco
>»Aco
Figure 14
about equal. When A and B are equal and a = 0, the bottoms of the
valleys become cusps, and the two envelopes are joined there to form
sinusoidal envelopes, as shown in Fig. 11.
In forced vibration near resonant frequency, if the system has a slight
amount of damping, the beat will die out, since one of its component
(1.6) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 23
vibrations is a transient. On the other hand, if the system is subject to
slight disturbances from various sources at all times, beats of various
amplitudes but the same beat frequency will be observed continually. The
important point is that since the transients have a frequency very near to
that of the steadystate vibration during their brief time of existence they
Can make their influence distinctly observable by forming beats with the
steadystate vibration.
(b) damped system
We consider next the steadystate response to a harmonic force of a
system with viscous damping. Let the equation of motion be
mx + ex + kx = F Q cos w f t (38)
The solution of this equation is known to be of the form 22
x(t) = C^ + C 2 e s * 1 + A cos (a> f t  a) (39)
in which A and a are determined by substituting (39) into (38):
— mco f 2 + k = (F /A) cos a
cco f = (f /A) sin a
coj f 2(cjc c )((x) f lcQ)
(40)
tan a =
—moo 2 + k 1 — (cOf/co)
(41)
^o A.
V(mw f 2 + kf + c 2 co/
\ CO 2 J \ C r 0) )
These relationships can be geometrically represented by a parabola, as
shown in Fig. 15. The validity of this representation, first given by C.
Runge, can be readily verified. By either the analytical expressions of (41)
or the graphical construction shown in Fig. 15, graphs of the magni
fication factor /l/(5,, and of the phase lag a as functions of co f jco and cjc c
can be plotted as shown in Fig. 16. They represent the socalled frequency
response of the system. A general discussion on the frequency response of
linear system is contained in Art. 1.11.
The maxima and minima of the curves in Fig. \6a are of practical
22 Although (38) can be solved by standard methods for solving differential equations
of this type, we employ the following intuitive reasoning. The solution consists of a
transient that must be the same as the solution of (14) and a steadystate motion that
must be of frequency o) f . The steady state, however, may or may not be in phase with
the force, so we introduce a phase difference a.
24
THEORY OF MECHANICAL VIBRATIONS
(1.7)
interest, as illustrated in Art. 1.15. For c/c c < 1/V2 all the curves possess
a maximum lying between frequency ratios of and 1 and a minimum
at (0, 1). The point (0, 1) becomes the maxima of all curves with damping
ratio greater than 1 / V2. Exercise 1 . 1 7 is devoted to the verification of these
statements.
The curves for the phase lag a, shown in Fig. 16b change gradually
from to 77 as co f increases. For different values of c\c c the curves first
diverge from the origin, converge at (1, tt/2), diverge again, and then
approach tt asymptotically.
5st ~^ >
1
■^^^ ^Parabola
1 \
■^a.
Figure 15
1.7 Complex Number Representation
The amplitude of the steadystate response / in (39) is assumed to be a
real number. By replacing it with a complex number, it can take into
account the phase difference a ; and we may write
x = Re (Xe^) = Re (f)
F cos (a t t = Re (F e i(O ' t )
and
Since (42) satisfies (38)
m ^ Re (0 + c i Re (£) + k Re (£) = Re (F e^
Since ra, c, and fc are real constants and t is a real variable,
Re {ml + c£ + *£) = Re (F e^)
(42)
(43)
Now if we change our "zero point" on the time scale and make the external
force F sin co f t instead of F cos co f t, it is evident that the steadystate
(1.7)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
25
IXI
S s t
1
I
c/c c
=
ft
c/c c
= ¥
1
ffl
c/c c
= _ 5"
c/c c
~ 2 "~
I
3
\
c/c
:= 1"
3
^v
1 2
(a)
26 THEORY OF MECHANICAL VIBRATIONS (1.7)
response of the system will not be altered in its physical character, although
its analytical expression is changed accordingly. In other words, the
motion is still of the same amplitude and frequency, and it maintains the
same phase difference with the force as before. Thus its analytical ex
pression is the same as the last term in (39), except that the cosine function
is changed into a sine function. Therefore the imaginary part of £ satisfies
Im (ml + ct + kQ = Im (F e ico ^)
By combining the two we obtain
ml + cl + kZ = F e iuj f l
(44)
mco/Xe^ + icwjie*** + Jt^ = F Q e iw f l
(45)
F F lk
A =
—ma) f 2 + k + icco f i co f\ 2 .(2 C( °f\ (46)
\co 1 \C C CO J
The absolute value of A, which we denote by the symbol A is then
(47)
^o
VCmay 2 + kf + cho f 2
and the argument of A, which is the phase difference between the force and
the displacement, is
ceo
a = arg (A) = tan" 1 f— —
— mo/ + k
or
tan a = ^— (48)
— mo)/ + k
These are identical to the results previously obtained.
The purpose of this analysis is to show that from the viewpoint of
solving problems it is not really necessary in complexnumber representa
tion to take the real part only. We do not have to make the differentiation
between x and £. and it is just as well if we write (38) as
mx + ex + kx = F e ia ^ (49)
with x — Xe iu)ft as the steadystate response. This is permissible if, and
only if, all the terms contained in equations such as (38) are sinusoidal
functions of the same frequency; because then the rotating vectors,
representing the various terms, maintain fixed positions relative to one
another as they rotate with the same circular frequency. Any additive
relationship existing among their projections must also hold vectorially
(1.7) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 27
among themselves. The vectors, which represent the inertia force mx, the
damping force ex, and the spring force kx, are shown in Fig. 17. Their
vectorial sum is the vector representing the external force. 23
It cannot be overemphasized that in applying mathematical tools to
solve a physical problem a set of "ground rules" is always agreed upon
beforehand by the users. Literal interpretation of symbols beyond what
the ground rules allow is meaningless. 24 In the present analysis it is
understood that we are interested only in getting the amplitude and phase
difference of the steadystate response of a linear system to an external
force which is a sinusoidal function of time. Under such circumstances, it
has been shown that with proper representation complexnumber algorism
Figure 17
can yield the correct results. No claim is being made that all physical
aspects of force and displacement can be adequately described by the
mathematical properties of complex numbers.
In the transient analysis we have shown that a damped oscillation can
be represented by the real part of a complex number
x(t) = Re (£) = Re (Xe iot )
It is rather simple to verify that the imaginary part of £ also satisfies (14).
Hence the complex number itself also satisfies
ml + ct + k£ = (50)
and, with proper understanding, we may dispense with the difference
23 The parallelogram law for the addition of coplanar vectors is equivalent to the
addition rule of complex numbers. Using mathematical terminology, one says that
coplanar vectors and complex numbers are isomorphic under addition laws.
24 Even the common practice of representing a scalar physical quantity by a real
number involves a prior agreement of units and method of measurement. It is meaning
less to say, for instance, that boiling water is 212/32 times as hot as ice.
28 THEORY OF MECHANICAL VIBRATIONS (1.8 J
between x and £. Our problem becomes that of looking for a solution of
(50) of the type £ = le iat , satisfying the initial condition specified by (23).
y i • C c x ~T" COJX
bo — A — X ~ l
At this moment some of the keener readers may recall that according
to the theory of differential equation a secondorder equation requires
two, not one, initial values for its complete solution whereas only one is
given by (23). There are several ways to explain this discrepancy, but all
are based on the fact that in our scheme of things the other solution of
(50), viz., £ = X'e tat (where a is the complex conjugate of a), can be
ignored, since, being a mirror image of the other in Argand's diagram,
it represents nothing that is not already included. Because we have only
one integration constant, K, to determine, we need only one initial value.
1.8 SteadyState Response to Periodic Forces
It is known that a periodic function of time can be represented by a
Fourier series; and under a wide range of conditions, which are usually
satisfied in a physical problem, the series converges to the value of the
function almost everywhere. 25
oo
F(t) = ^ ( a n cos nco ft + b n sin noj f t)
(51)
00 00
= 2 /. cos (nco,t  pj = J Re (<f>„e<"^)
»=0 »=0
where T = lirjcOf is the period of the function and for each n the constants
a n , b n , f n , f$„, and <j> n are interrelated in the usual manner. It is well
known that the coefficients a n and b n are given by
«0 = r\ F W dt ^0 =
1 JT/2
2 f T/2
a n =  F(t) cos nco f t dt (52)
1 JT/2
n>0
2 C TI2
b n = — F(t) sin nw f t dt
T JT/2
25 See standard mathematics text about Dirichlet's conditions in Fourierseries
representation. The term "almost everywhere" has precise mathematical meaning with
which we need not be concerned here. Sufficient is it to say that the series will converge
to the function at all points of continuity.
(1.8) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 29
A periodic force, when expressed by a convergent Fourier series, is thus
the sum of sine and cosine functions. By the principle of superposition,
which is discussed fully later, the response of a linear system to a periodic
force can be expressed as an infinite series whose terms are the responses to
the terms in the Fourier series, provided that both series are convergent.
Utilizing the results previously obtained, such as (33), (39), and (41), we see
that the steadystate response to a periodic force for an undamped spring
mass system is
1 °° f
x = t 2 : rTT2 cos ("<V  Pn) (53)
k n=o 1 — {ncOflcoY
and for a damped springmass system,
x = 2 y fn cos (nco f t  n  a B )
kn=o // n 2 co 2 \ 2 (2cncoA 2 (54)
n 2 co f 2 \ 2 1 2c nco f \
co 2 ) \c r co /
where
(2clc c )(nco f lco)
^ = tan ~\ — 7 — r^
1 — {ncOfjcoY
Both expressions are a Fourier series; the motion they represent is
therefore periodic and has the same period as that of the force. In com
paring the coefficients of these series with those of (51), it is seen that unless
co is an integral multiple of co f the convergence of (53) and (54) is assured.
As a matter of fact, the series representing the motion converges much
faster than that representing the force, and one need not use too many
terms of (51) to obtain approximate answers for x in (53) and (54). 26
The studying of a periodic phenomenon by resolving it into sinusoidal
components is called harmonic analysis. Besides the formal mathematical
operations expressed in (52), there is a mechanical or electronic device,
called harmonic analyzer or wave analyzer, which will "measure" the
coefficients/^ (and sometimes also /?„) of a periodic phenomenon. Numeri
cal procedure for computing the coefficients in the Fourier series, repre
senting a physical phenomenon based on a finite number of measurements
taken on the phenomenon, is also available. 27
26 Note also that a n * n as n > co.
27 For details see Scarborough, Numerical Analysis, Johns Hopkins Press, Baltimore,
1955, 3rd edition, pp. 477494.
30 THEORY OF MECHANICAL VIBRATIONS (1.9)
1.9 Work Done by External Forces
and Energy Dissipation in Vibratory Systems
The work done in a time period T by a force F(t) acting on a body
moving with velocity x(t) in the direction of the force is given by the
integral
W= \ T F(t)x{t)dt (55)
Jo
If both the force and the displacement are sinusoidal functions of time
having the same frequency co f and a phase difference of a, then the work
done per cycle of the motion is
d
r2ir/(o f
W = F cos co f t ^ A cos (co f t — a) dt
Jo
= 7rF A sin a
Thus it is seen that work done per cycle is not only proportional to the
amplitudes of the force and the displacement but also to the sine of phase
difference. In addition, there is the constant of proportionality 77. In
steadystate forced vibration of an undamped system a is or 77, except
at resonance. The work done per cycle is therefore zero, and no energy
is accumulated. At resonance, as discussed in Art. 1.6, a becomes 77/2,
and there is a net amount of work gained by the system for each cycle
proportional to the amplitude of that cycle. Since the system contains no
mechanism for energy dissipation, the amplitude must build up each cycle
by an equal amount. 28 This accounts for the straightline envelopes in
Fig. 12. For the steadystate vibration of a damped system sin a does
not vanish, the net work performed by F is dissipated by the damping
mechanism in each cycle, and a constant amplitude is maintained. Since
the damping force is the only force that can dissipate mechanical energy.
the work done by the damping force per cycle must be equal to that by
the external force. Hence
ttF\X\ sin a = cx(t)x(t) dt
Jo
PT
V 2 sin 2 (co f t <x)dt = ttcA
Jo
= C H
or
F c(D }
A sin a
(57)
(58)
28 The truth of this statement lies with the fact that the total energy of the system, E,
is proportional to the square of the amplitude. E oc A 2 and (dE/dt) oc /.; hence,
d\)\\dt = constant.
(1.10) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 31
By the use of (47) and (48) it can be readily verified that the foregoing
relationship is correct. For systems with a small amount of damping A
is almost maximum at a = 77/2. (See Fig. 16 and Exercise 1.17.) The
damping of the system can thus be measured by exciting the system to
vibrate with maximum steady amplitude and compute c from the approxi
mate relationship
^0
1 dT
1 dV
P
=
K =:
C :== —
xx dt
xx dt
X 2
(59)
^/Hmax
The energy relationship also serves as a convenient means for setting
up the differential equation of motion, especially when the system contains
a number of constraint forces, which do no work and can thus be ignored
in the analysis. An example is given in Art. 1.12. The basic energy equation
is
d d dW
dt dt dt
in which P is the power of the dissipative force and W is the work done
by external forces. The parameters of a singledegreefreedom system can
then be defined by
^ \dW
and F =
x dt
Depending upon the coordinate variable x chosen, the quantities m, k, c,
and F, may or may not have their usual dimensions.
1.10 Response of Linear Systems to a
General External Force — Superposition Theorem
The systems discussed so far are called linear systems because their
equations of motion are linear differential equations having the form
L(x) = F(t) (60)
where L is a linear differential operator.
L = a ° (0 a¥ + a±(t) aW^'" an ~ l(t) dt + aM
For the systems studied n = 2 and the a's are constants instead of
functions of time t. The following discussion is, however, pertinent to
general linear systems.
An important property of linear systems is that they follow the principle
32 THEORY OF MECHANICAL VIBRATIONS (1.10)
of superposition. This property is derived from the fact that linear opera
tions are distributive ; that is,
L(x 1 + x 2 ) = L(^) + l(x 2 )
Thus if x = x ± (t) is a solution of L(x) = F ± (t), satisfying the initial con
dition x(0) = x 10 and x(0) = x 10 , and x = x 2 (t) is a solution of L(x) =
F 2 (t), satisfying the initial condition x(0) = x 20 and x(0) = i 20 , then
x = ax^t) + /fo 2 (0 is a solution of the differential equation 29
l(x) = aF x (r) + /JF 2 (0
and satisfies the initial condition 30
x(0) = ax 10 + fix 20 x(0) = ai 10 + f3x 20
in which a and /? are arbitrary constants.
Let the equation of motion of a linear system under the influence of an
external force F(t) be
L(s) = F(t) (61)
and let x = u(t) be the solution of the differential equation
l(x) = S(t) (62)
satisfying the initial condition x = 0, x = 0, in which J~(r) is a unit step
function defined as
J"(0 = for t <
J"(0 =1 for t >
This function «(/) is called the indicia! response or indicial admittance of the
system. It gives the displacement of the system, if it is initially resting at
its equilibrium position and is suddenly subjected to a constant force of
unit magnitude applied at t = 0.
From the principle of superposition it can then be seen that the response
of a system that is initially at rest and then subject to a constant force A
29 This theorem was also implied in deriving (53).
30 We assume that the differential equation is of the second order. Otherwise, the
initial condition shall contain the values of derivatives up to (/? — l)th order. This
assumption does not affect the generality of subsequent results.
(1.10) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 33
suddenly applied at t = and to another force B suddenly applied at
/ = ris
x(t) = Au(t) + Bu{t  t) t > r > (63) 31
We assume here that the system is time invariant; that is, the physical
property of the system does not vary with / or the operator L does not
contain / explicitly. Otherwise u is a function of two variables u(t, r)
where r is the time when the force is applied and t is the time when the
response is measured. u(t) then stands for u(t, 0). Most physical systems
whose properties vary gradually with time because of aging of components
*~t
Figure 18
or changes in the environment have become of interest. Such gradual
changes, however, do not affect their shorttime behavior.
A general timedependent force F(t), which is applied to a system at
time t = 0, can be considered as the sum of finite and "infinitesimal"
step functions, as shown in Fig. 18. Assuming, for the time being, that
31 The method of defining u(t) previously given leaves it undefined for t < 0, so that
(63) is valid only for t > r > 0. However, if we extend out domain of definition by
assigning
«(/) = for t <
then (63) is valid for all values of t and t > 0. For what follows this extension is not
necessary.
34 THEORY OF MECHANICAL VIBRATIONS (110)
F(t) is continuous and differentiable for t > 0, we may write
F(t) = F S(t) + dFS(t  r)
(64)
F S(t) + F'(t) T(r  t) dr
By the principle of superposition, if this force is applied to a system that is
initially at rest at its equilibrium position, the response will be
f V'(r)«(r 
Jo
x(t) = F u(t) + F'(r)u(t  t) </t (65)
Jo
in which u(t — t) is the response of the system at t due to a unit step force
applied at r. If «(?) is continuous at f = and differentiable for t > 0, as
in most problems, we can integrate (65) by parts to yield
M J
x{t) = F u(t) + [F(r)u(t  t)] I" 1  F(t)  u(t  r) dr
l T==0 Jo AT
Now w(0) = and (djdr)u{t — r) = — w'(' — t); hence
x(0 = fV(r) W '(*  t) £/t (66)
^o
Since the function F appears under an integral sign, the requirements of
continuity and differentiability of F are not necessary insofar as (66) is
concerned. 32 Although these requirements were made in deriving (66),
the same results can be directly obtained by another line of reasoning.
Consider a timeinvariant linear system that is in equilibrium and at rest
prior to t = 0. At / = the system receives an impulse of force of unit
magnitude. 33 The effect of this unit impulse is to produce an initial
velocity x = \\m. With this initial velocity the system is now set into
32 If the integral in (65) is considered a Stieljes integral, the requirement on F is not
very stringent. At any rate, in an actual physical problem Fhas at most a finite number
of discontinuities and points where F' does not exist; the mathematical difficulties can
easily be overcome by adding the term AFw(r — r^, corresponding to a jump discon
tinuity AFat t = t 1} to (65) and splitting the integral to cover ranges where F'(t) exists.
After integration by parts, the result is still (66), since the terms corresponding to the
discontinuities in F cancel out just as F u(t) term does.
33 One may speak of a unit impulse of force at t = as a force described by the Dirac
function b(t). Although pure mathematicians are still somewhat uneasy about a precise
definition of such a function, applied mathematicians have accepted it with little qualm.
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
35
(1.10)
motion. Let the response of the system be such that its displacement is
given by the function h(t). That is to say
x = h(t)
satisfies
l(x) =
and
1
x„ =
m
pn
^dr
Figure 19
In a more general discussion we must assume that L is of rath order.
A unit impulse will then produce the initial condition
X(\ — Xt\ —
(n2) _
and
~(nl) _
in which a is the first coefficient in L. For the convenience of visualization
we assume that L is secondorder without losing important generalization.
Figure 19 shows that the action of a force F(t) on a system may be
considered as that of a series of elemental impulses F(r) dr applied succes
sively to the system in the interval < r < t. According to the principle of
superposition, the response of the system at t is the sum of the responses to
these impulses. Hence
x(t)
Jo
F(r)h(t  t) dr
(67)
36 THEORY OF MECHANICAL VIBRATIONS (110)
In comparing (67) with (66), we see that 34
h{t) = u'{t)
Thus the displacement response of a linear system to a unit impulse is
equal to its velocity response to a unit step force. Either way we define it,
the function u'(t) is called the impulse response of the system.
In all these derivations it was assumed that at t = the initial condition
of the system is x = and x = 0. If this is not the case, it is not difficult
to see that the solution of the problem merely requires additional terms
corresponding to the nonvanishing values of x and x .
Let us now find the indicial response and the impulse response of a
linear damped system. For u(t) we solve the equation
mx + ex + kx = 1 t > (68)
together with the initial condition x = 0, x = 0. The general solution
of (68) is evidently obtained by adding the particular integral \\k to (21).
x = u(t) = Re (Xe iat ) +  (69)
k
It appears that in what follows we assume cjc c < 1 . However, according
to Exercise 1.11, the results are valid for all damping ratios.
or
Since from (27)
or
x = Re (A) +  =
K
x = Re (iol) = = Im (aX)
Re (a) Im (A) = Re (A) Im (a)
ICOC
X = (1  /tan (5) = 
(70)
k kcosd (71)
u\t) = Re (ioXe iat ) = Im (oXe iat )
34 The equality of these two functions cannot really be established merely by com
paring (66) and (67) because / appears both in the integrands and in the limits. Never
theless, these functions can be shown to be identical. To prove this fact, utilize the
formula to be proved in Exercise 1.32 and remember that F(r) can be any arbitrary
function.
(1.10) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
Utilizing (22), (27), and (71), we have
CO 2
u(t) =—  e< cl2m)t sin co c t
co„k
37
(72)
Derivations for the same relationships by the classical method are as
follows (see also Exercise 1.24):
x = u{t) = Ce~ {cl2m)t cos (aj c t  a) + 
(69a)
M (0) = C cos a = 
u\t)= Ce {cl2m)t
_2m
cos (co c t — a) + co c sin (co c J — a)
u'(0) = — cos a = ca c sin a
2m
division of the equation by w and the application of the relations in (27)
yield
1 CO
a = 5 C =  =  —
k cos a kw n
oy
koj n
■(cl2m)t
u(i) = U 1  — e  (cl2m)t cos (oj c t  S)
k\ co c
(69b)
Equation (72) can be obtained much more readily if we solve for the
problem of free vibration due to an initial velocity x = 1/m. 35
In that case we take the general solution (21):
As
Also as
and
x = h(t) = Re (Xe iat )
x = so Re (X) =
^ = Re (iaX) = Im (gX) = Im (A) Re (a) = —
m
Re (o  ) = co c Im (X) =
35 On the other hand, the reasoning we used in deriving (67) is not mathematically very
rigorous, although such rigor can be had by refining some of the arguments.
38 THEORY OF MECHANICAL VIBRATIONS (1.10)
SO
AL
moj c
and
A(0 = Re — e iat ) = m'(0 (73)
\ moj r J
It can easily be verified that (73) is the same as (72). The foregoing results
contain as a special case the indicial response and impulse response of an
undamped system. For such a system let
c = b =
(74)
sin cot
CO =
oj c =
= a
c =
Thus (69) and (72) become
u(t) =
■fi
■ cos cot)
u'(t) =
CO
 — sin o)t =
k
1
mco
sin cot
Vmk
Substituting (72) into (66), we have the solution for forced vibration of a
damped system, which is initially resting at its equilibrium position.
If the system has an initial displacement and velocity, complementary
solutions (19) with the constants determined in (23) must be added. The
complete solution is therefore
x(t) = e«l**»(x cos co c t + Cc± « + CMX ° sin co c t)
\ OJ r C r J
co c c c
P CO 2
+ —7 e {cl2mKt  T) sin co c (t  t)F(t) dr (75)
Jo co c k
For an undamped springmass system, with c = and co c = to, the
solution becomes
x(t) = x cos cot H —  sin cot + — sin co(t — r)F(r) dr (76)
w Jo k
In summary, the particular solution satisfying a linear differential
equation with constant coefficients and a set of initial values
\_(x) = F(t) when / = x = x x = i
etc., is made up of two parts; one part represents the motion 36 which
36 We use the words "motion" and "force" merely to fit the physical phenomenon
under discussion. The theorem is, however, strictly mathematical and can have other
applications.
(1.11)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
39
would prevail if F = 0, and the other part represents the motion caused
by the force F{t) acting on the system subsequent to t = 0. 37 In contrast,
the classical method illustrated in Arts. 1.7 and 1.9 obtains the general
solution to the differential equation first; this consists of a complementary
function, representing the transients, and a particular integral, represent
ing the steadystate motion. 38 The particular solution, when needed, is
then obtained by determining the constants in the complementary function
to satisfy the initial conditions. When the applied force is periodic, and
only the longterm behavior of the system is of interest, this method of
solution is a natural one. Conversely, when the applied force is of short
duration and only the shortterm behavior of the system is wanted, the
method described in this article is more natural. Nevertheless, as far as
mathematics is concerned, either method can be used in either case.
1.11 SignalResponse Relation
of Linear Systems in General
The problem we have studied in the last few articles was formulated in a
restricted way. We sought the forcedisplacement relationship in a spring
massdamper system of a certain arrangement. The essential part of the
analyses has, however, much more general applications.
■w/aw/W/WMwy/^^^^^
»wv\w
Q
^/////////f///^////////////////f//////////////A
'/77/W//W///7//
^A
Figure 20
Let us visualize a springmassdamper system enclosed in a "black box,"
as shown in Fig. 20. The three weightless rods, A, B, and C, which are
attached to the components of the system, protrude through the walls of
the box and can either slide without friction or be locked to the wall. The
problem investigated was, "What kind of displacement will be produced
37 In this approach the condition of the system prior to / = is not considered at all.
38 The terms "transient" and "steadystate motion" are devoid of physical significance
when the applied force is not aperiodic function of time, although the two parts of the
solution are commonly so designated.
40
THEORY OF MECHANICAL VIBRATIONS
(1.11)
in A by a force applied at A, provided that B and C are locked to the box,
which is held fixed?" In other words, "How does the black box transform
a force applied at A into a displacement of A ?" Evidently, this is not the
only behavior of the box that could be of interest. For instance, one may
investigate, "What is the force transmitted to the foundation, to which the
box is attached?" Or one may change the operating condition of the
system by loosening the lock on B and moving B by some external means,
and then ask, "What will be the motion transmitted to A ?" We can also
ask the same questions about a more complex system, such as that shown
in Fig. 21. In all of these cases an external disturbance or excitation,
such as a force, an impulse, or a displacement, is imparted to the system
mms^z^^&m^^mimsmmzEEzzzzzzzzzzzz^^^^ EZEz^z^Ezzmzzzzz
Figure 21
at a certain place, and we look for a certain resulting change in the system
that interests us. This excitation, transient or periodic, is called a signal,
and the change in the system that interests us is called the response to the
signal. Clearly, the designation of a particular change as the response is
purely a subjective matter. What we wish to emphasize here is that many of
the analytical results obtained in the preceding articles can be immediately
generalized to cover the allsignalresponse relationships in linear systems.
(A) TRANSIENT RESPONSE — INDICIAL RESPONSE
Consider the superposition theorem described by (66) and (67). Since
in the derivation of these equations no reference is made to the exact make
up of the systems or to the nature of the signal and the response, it is
evident that they apply to all linear systems with timeinvariant components.
Take, for instance, the question of the force transmitted to the foundation
of the box of the system in Fig. 20. If F(t) is the force applied at A and
(1.11) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 41
/(/) is the force transmitted to the foundation, the problem can be formu
lated by the following set of differential equations:
mx + ex + kx = F(t) ,
ex + kx=f(t)
Although it is possible to eliminate the variable x in the foregoing set of
equations to obtain a single equation in the standard form of
mf + cf + kf= cF+ kF= G{t) (78)
it is not necessary to do so. 39 According to our previous reasoning, if
u f (t) is the force transmitted to the foundation when a unit step force is
applied at A, the corresponding force transmitted with F(t) is given by
fit) = P »/
Jo
(t  r)F(r) dr (79)
provided that the system is initially at rest and in equilibrium.
To evaluate u f (t), we can first evaluate u x {t), which is the solution to
the equation
mx + ex + kx = S(t) with x = and x —
and then utilize the relationship
cu x {t) + ku x (t) = u f (t)
If we are to analyze the same problem for the system in Fig. 21, the
indicial response function u f (t) will be a more complicated one and will
have to be obtained by methods discussed in later chapters; but when it is
found the rest of the procedures will be the same.
The most important lesson to be learned from this discussion is that the
dynamic property of any linear system is completely specified by its
indicial or its impulse response. These functions describe how a system
behaves under any given excitation or how it will change any given input
signal into an output. In short, they describe the system just as completely
as the differential equation itself. Furthermore, for many actual systems,
because of the uncertainties in the measurement of such parameters as
masses and spring constants, the differential equations themselves are
likely to be less reliable than the indicial response functions when the
latter can be measured directly.
(B) STEADYSTATE RESPONSE — FREQUENCY RESPONSE
Another way of describing the dynamic property of a linear system is
by its steadystate response to sinusoidal inputs. This description is
39 See also Art. 1.17.
42 THEORY OF MECHANICAL VIBRATIONS Oll)
furnished by the socalled frequency response of the system, that is, the
amplitude and phase difference (lag) of the response as functions of the
frequency. Take, for instance, the system shown in Fig. 20. Let the
signal be in the form of a sinusoidal motion imposed on B, and let the
motion of A be considered as the response, with C clamped to the box.
The differential equation relating the displacements of A and B is
mx a + cx a + k(x a — x b ) =
Upon letting
x h = \).,\ COS CO f t
and
x a = \?. a \ cos (m f t  x)
a comparison with (38) shows that the amplitude ratio / a //. 6  and phase
lag a are the same as the magnification factor 40 \/.\ld st and angle a defined
in (41).
It is more convenient, however, to use complexnumber representation
to describe the signal to response relation so that both the amplitude ratio,
generally called the gain, and the phase lag are combined in one expression.
According to (46), the frequency response in this case is given by the com
plex number
§ 'a
K
1
~() 2 +
£0 '
in which / a and ). h are then the complex amplitudes of A and B. respectively.
Consider now the relation between the force transmitted to the founda
tion /and the force applied to A, that is. F. According to (78), if we let
f=& m > % and F=F e^
we have
/ icco, + k
F k — mcof + ieco f
= T r (ico f ) (80)
The absolute value represented by this ratio, or the gain, is called trans
missibilitv.
f " N [C <" (81)
"i _ te)T+ p5r)'
V CO 1 J V c c CO '
40 In the present connection the name of this factor gives a more graphical description
to its physical nature.
(1.11) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
4.0
43
Figure 22
This ratio is plotted in Fig. 22 and is of practical interest in the design
of supporting structures for machines susceptible to vibration.
(C) TRANSFER FUNCTION AND TRANSFER LOCUS
In general, if X r is the complex amplitude of the response and X s is that
of the signal, the ratio between the two is a complex function of variable
io) f , which describes the frequency response of the system
l r = T{ko f )X s (82)
The two examples show that the function T is a ratio of two poly
nomials of ico f with real coefficients, which are the coefficients in the
44 THEORY OF MECHANICAL VIBRATIONS (1.11)
differential equation. Although in our discussion ioj f is always imaginary,
to study the analytical properties of this function, hence of the differential
equation itself, it is sometimes advantageous to extend its domain of
definition to allow complex argument. Because T is a rational function
this extension, by replacing the pure imaginary argument, uo t with a com
plex argument s, produces function values of T which "blend smoothly" 41
f' mo) f 2 + icw f + k
Im [T(ico f )]
Figure 23a
into the values in its restricted domain of definition. The utility of the
function T(s), called the transfer function, is connected with the theory
of Laplace transformation, a powerful tool in analyzing linear systems,
which, unfortunately, cannot be discussed at length here. It is mentioned
only to introduce the reader into a casual but logical acquaintance with a
41 Without entering into the theory of analytical continuation of analytic functions,
this is the best we can do to describe this process. The statement is intended merely to
convey some rough idea of a process which is not of primary concern to us here.
(1.11)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
45
This
subject that, it is hoped, he will study more intensively elsewhere
remark applies also to the rest of the discussion under this heading.
To represent the function T(ko f ) graphically, we have been plotting its
absolute value (or the gain) and its argument (or the phase lag) as functions
of o) f . This method results in two curves. There is another method of
T(s) =
ms z + cs + k
0.2 0.4 0.6 0..
Figure 23 b
1.0 1.2 1.
1.6 1.
graphical representation which crossplots the two to give a locus of T in
Argand's diagram with co f as a parameter. Such a plot is called the
Nyquist diagram or Nyquist locus. The Nyquist locus, representing the
frequency response of a simple massspringdamper system, is shown in
Fig. 23a. Figure 23& assembles a number of such loci with different values
of cjc c . Consider T(s) as a function of a complex variable s, the Nyquist
locus is then the "map" of the imaginary axis of the splane on the T
plane.
46 THEORY OF MECHANICAL VIBRATION (1.11)
The generic term transfer locus is used to include the Nyquist locus, T,
the inverse Nyquist locus IjT, and the overall transfer locus Tj(\ + T).
All are useful tools in the analysis and design of linear control systems.
(d) relation between the indicial response
and the frequency response of a linear system
We have seen that both the indicial response function and the transfer
function can be derived from the differential equation describing the
system. We have also shown that the indicial response by itself describes
adequately the dynamic properties of a system. We now proceed to show
how the indicial response function is directly related to the transfer function.
This relationship, when established, serves two purposes. Analytically, it
proves that the frequency response also gives an adequate description of
the system. Experimentally, it furnishes a better way of determining the
dynamic characteristics of a system because the measurements of the
frequency response, which are those on a steadystate phenomenon, are
easier to make than those on a transient one.
It is necessary here to digress into a mathematical formula to be used
later. Any standard textbook 42 on advanced calculus contains the evalua
tion of an improper integral
!*% = *
6 2
f
Jo
Upon replacing 6 by co f t, it is not difficult to see that
f x sin co f t ,
f dco f =\
Jo c>,
/ =
The step function _T(f) can then be expressed in terms of this improper
integral as
1 1 f °° sin (oJ , , M ^m*
T(0 =  +  f dco f (83) 13
2 77 JO CO f
42 See, for instance, Sokolnikoff, Advanced Calculus, McGrawHill, New York, 1939,
pp. 361, 362.
43 Strictly speaking, this equation is not valid at one point, viz., t = 0, when the
lefthand side is defined as 1 and the righthand side is \. This difference is, however,
of no consequence here.
(1.11) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 47
The indicial response u(t) to a linear system represented by the differen
tial operator L can thus be considered as the solution to the equation
(/ x 1 1 f 00 sin ay J
L 77 J0 COf
(84)
It must be pointed out that replacing (62) by (84) really involves a
change in physical considerations as well as in the mathematical descrip
tion of the problem. The step function S(t) was previously chosen to
characterize a nonperiodic excitation. Although the function value was
also defined for t < 0, its definition for negative values of t is not essential
because our analysis covers only the time period beginning at t = when
w (0) = w'(0) = . . . = ^"^(O) = 0. On the other hand, physical reason
ing clearly shows that if we extend the time axis of analysis backward to
t = co the fact that S(t) = for all negative /'s means that the system
has been completely free from excitation since time immemorial and
consequently it must be in a completely quiescent state at t = 0. 44 There
fore, by utilizing the definition of a unit step function for negative values
of /, we have altered our consideration of the indicial response from that
due to a nonperiodic excitation to that due to a periodic excitation of
infinite period, which, according to (83), is the sum of sinusoidal functions
of all frequencies. Each frequency contributes an amount equal to dco f /7TCo f ,
and the contribution of zero frequency is the constant h
Let the transfer function associated with the differential operator L be
T(s) and the real and imaginary parts of the frequency response function
T(ko f ) be R and /, respectively,
T(ico,) = R(co f ) + U(o),)
Equation (82) can then be written as
l/^ = [R( j f ) + /Y( W/ )] A/^
Let k s = 1, X s e lMft = cos co,t + i sin co f t.
Re (X r e iu} ^) = R(oj f ) cos co,t — I(co f ) sin co,t
Im (h r e 1UJft ) = R(co f ) sin co,t + 1(a),) cos o f t
According to the discussion in Art. 1.7, the righthand sides of the
last two equations are the responses to signals equal to cos co,t and sin
44 In essence, we are assuming that the system is stable and has energy dissipation
mechanism.
48 THEORY OF MECHANICAL VIBRATIONS (1.1 1)
(o f t, respectively. Hence we have the following signaltoresponse relation
ship:
Signal or Excitation Response
i = i cos 0/ i [R(0) x cos Ot] = iR(0)
sin co f t R(oj f ) sin oj f t + I(oj f ) cos oj f t
T(t) u(t)
The solution to (84) can thus be obtained through superposition:
u(t) = \R(0) +  [/tfw,) sin co f t + /(a),) cos co f t] — (85)
7T Jo 0)^
If the resulting improper integral converges uniformly with respect to ?, we
may differentiate (85) with respect to t to obtain
1 f 00
u'(t) = — [R(oi f ) cos oy — I(co f ) sin w^] ^/c/^ (86)
77" JO
Before we proceed, it must be realized that the operations leading to
(85) and (86) are strictly formal; that is to say, there is a number of
mathematical questions that must be settled before we can accept the
results as legitimate. The subject belongs to the theory of Fourier trans
forms, which is not within the scope of this book. We can, however, deal
with the more essential points here.
Let us introduce the symbol L _1 to denote the operation that yields the
particular integral of a linear differential equation:
L(*) = F(t)
x = LiF(0
This inverse operation can be carried out by a number of methods,
including such procedures as Lagrange's variation of parameters. We do
not have to be concerned with the exact method used, except that essentially
it involves integration processes. The solution to (84) should be of the
form
r, , J l \ l , i f 00 sin ay ,
u(t) = L 1  1 +  Li L da> f
\2! rr Jo tO f
whereas (85) is in reality
/1\ 1 f 00 , JsmG) f t\ ,
Since L 1 involves integration with respect to /, the validity of (85) depends
upon whether the order of the two integration processes can be reversed.
(1.11) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 49
For the integral in question the sufficient condition for the legitimacy of
order reversal is that the integral should be uniformly convergent for any
finite ranges of t, — oo<a<t<b<co. Upon examining the inte
grand, we see that uniform convergence is assured if I(co f )jcOf has a finite
limit at co f = 0, I(co f ) approaches at co f = oo, and the absolute value of
T(ico f ) is always finite. The first two conditions are satisfied when I(co f ) is
the imaginary part of a transfer function. (See Exercise 1.34.) The last
condition is satisfied when the system is damped and has no resonance
condition.
However, with more sophisticated mathematical tools, (85) and (86)
can be derived without the requirements that T(ico f ) be bounded and that
the integrals converge uniformly. Instead, these integrals are interpreted
as the limits of a sequence of convergent integrals containing a parameter
a as a approaches 0. Physically, the procedure is equivalent to introducing
an extra damping force so that the resonance condition cannot take place
and then letting this force approach zero in the solutions obtained. If the
limiting integral converges to a socalled Cauchy's principal value, the
result is then valid.
Returning now to (85) and (86), we observe that u(t) and u'(t) should be
identically zero for / < 0. Hence, if Ms a positive number
= «(— t) =  R(0) +  [— R(co f ) sin co f t + I(co f ) cos co f t] — 
2 77 JO 0) f
Combining this equation with (85), by addition and by subtraction we
have
u(t) = R(0) +  I(oj f ) cos co f t — (t > 0)
77 JO (Of
or
u(t) =  ( R(w f ) sin co J ^ ) (t > 0) (87)
77 JO \ (O f J
Similarly, u'{—t) = 0, if t is a positive number.
1 f 00
=  [R(co f ) cos co J + I(co f ) sin co f t] dco f (t > 0)
77 JO
Together with (86), this relation gives
2 f 00
u'(t) = — R(co f ) cos co f t dcOf
77 JO
(88)
2 poo
= I(cQf) sin co ft dcOf (t > 0)
77 JO
50 THEORY OF MECHANICAL VIBRATIONS (1.1 1)
The expressions (87) and (88) are valid only for t > 0. When the
transfer functions are determined from the frequency response measured
experimentally, the integrals in these expressions can be evaluated by a
numerical procedure or by mechanical integrating devices. 45 When the
transfer function is known analytically, these integrals cannot be con
veniently evaluated as they stand. It is usually necessary to carry out
the integration process in the complex plane by the following trans
formation.
First, let us write (86), which is valid for all t, as
If 00
u'(t) =  Re [T(ico f y a ' t \ d<o f (89)
7T JO
Now consider the function
T(ico f )e^ = T(ico f )(l + ia> t t + ^ + ( ^ r* • • •)
(90)
in which T(ico f ) contains only real constants, except for the argument ko f
itself. Since i and co f enter together as a product in this function, its real
part is an even function of a> f and its imaginary part is an odd function
of co f . Hence
f °°Re [T{io) f )e im f l ] dco f = I Re [T(io) f )e i<0 ^\ da> f (91)
Jo J co
and
f °°Im [Tiico^e^] dco f =  f Im [T(ia> f )e ito ' t \ dco f (92)
Jo J  oo
We can then write (89) as
1 f 00
u\t) =  Re [T(iQ) f )e %m ^] dco f
77 Jo
o
Re [T(iw f )e iai S] dco f (93)
2ttJ
2ttJ
[T(ico f )e %m ' t ] dco f
The sign "Re" was dropped in the last integral, since its imaginary part
will vanish because of (92). Replacing the dummy variable ico f in (93) by s
puts the integration in the complex plane, and we have
w'(0=^. P°° T(s)e st ds (94)
Z777 Jioo
45 "Determination of Transient Response from Frequency Response," by A. Leonhard.
ASME Trans. Vol. 76, 1954, p. 12151236. See also discussion of the paper by A.
Tustin.
(1.11) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 51
This is an important relationship in operational calculus. It states that
the impulse response is the inverse Laplace transform of the transfer
function, or that the Laplace transform of the impulse response is the
transfer function. Ordinarily, the relationship is written
i rc+io
.77/ Jc—icc
u'(t) = — T(s)e st cis (95)
2.7TI Jc—icc
The constant c allows a shift in the path of integration from the imaginary
axis to another line parallel to the imaginary axis without changing the
value of the integral. 46 Such shifts may facilitate the evaluation of the
integral and avoid the mathematical difficulty when the system has
resonance conditions and T(ico f ) becomes infinite for certain values of
ico f . Similarly, (85) may be transformed to an integral in the complex
plane as
1 (*c+i co e st
u(t) = — T(s)  ds (96)
■^TTl J c — ico S
This integral is associated with the name of Bromwich.
As a practical matter, let it be pointed out that there are extensive
tables available which give the pairing of functions of t with functions
of s through Laplace transformation or its inverse. These tables, together
with the linear characteristics of the operation, make it unnecessary to
carry out actual integration processes for the evaluation of these integrals.
The primary purpose of this discussion is to show that there is a direct
connection between the steadystate response and the transient response.
In the process of showing this connection, we have taken a peek into the
subject of operational calculus, which is the most powerful tool for
transient analyses of linear systems. This subject is, however, too large
and important to be included in this book. Moreover, from the viewpoint
of applications, elaborate analysis of transient phenomena is seldom made
in connection with mechanical vibrations, as it is with control systems and
instrumentations, which form courses of study by themselves.
In concluding this article let us make one more generalization. At the
beginning we chose as a model a signaltoresponse relationship governed
by an equation of the type
L(z r ) = x s (97)
in which L is a differential operator with constant coefficients. There is
46 The theory of functions of a complex variable has precise things to say about how
much of a shift is permissible.
52 THEORY OF MECHANICAL VIBRATIONS (1.12)
nothing really to prevent us from dealing in exactly the same way with the
equation
l ± (x r ) = L 2 (x s ) (98)
or with a set of n linear equations in the general form of
IW= L i( x s) i= 1,2, •••,/!
3=1
with one of the x's at the lefthand side chosen as the response. In other
words, we have never restricted our reasoning to the case of singledegree
freedom systems. Consequently, with minor modifications, the results
obtained can be applied immediately to systems with multiple degrees of
freedom.
SECTION B. METHODS AND APPLICATIONS
1.12 Examples of Linear Vibratory Systems
with a Single Degree of Freedom
It has been pointed out that the systems shown pictorially in Figs. 1
and 5 are merely models of a class of physical systems whose outward
appearances may be quite different. In this article we give a few such
examples. 47 For this purpose it is necessary that we set up only the
differential equations governing the motions of these systems and show
that they are of the form of (2) or (14). The solutions to these equations
have already been studied.
(a) torsional vibration of a disk on an elastic shaft
Let an elastic shaft be fixed at one end and carry a disk of moment of
inertia /at the other end, as shown in Fig. 24. Assume that the mass of the
shaft is negligible. The configuration of the system can then be described
by the angle of twist 6. Because the shaft is elastic the torque exerted by
the shaft on the disk must be proportional but opposite in direction to the
angle 6. Hence
Id = k6 or Id + kd =
where k is the torsional stiffness of the shaft.
(b) oscillation of a liquid
in a utube of uniform inside diameter
Let the Utube shown in Fig. 25 be in a vertical plane and filled partially
with an incompressible liquid. The liquid levels at the two sides of the
47 For other examples see Arts. 1.15 and 1.16.
(1.12) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 53
tube may be set into an oscillation by momentarily applying a pressure to
one side of the tube and then releasing the pressure. Let 2x be the differ
ence in the levels of the two sides. The potential energy of the system is
equal to that of elevating a liquid column of length z to a height x. Hence
V = pgAx 2
in which p is the density of the liquid and A is the crosssectional area of
the tube. The kinetic energy of the system can be seen to be
T = ipALx 2
mwm.
c_>
T
X
i
V
Figure 24
Figure 25
in which L is the total length of the filled tube. If the viscous and capillary
forces are neglected, we have
i
(T + V) =
at
or
Lx + 2gx =
If the viscosity has to be taken into account and a laminar flow can be
assumed, the viscous force per unit length can be obtained from the well
known HagenPoiseuille law as
dF Sjux
~dl = ~i r
in which /u is the dynamic viscosity of the liquid and r is the inside radius
of the tube. The differential equation is then
8Ljux
pALx +
+ IpgAx =
54
or
THEORY OF MECHANICAL VIBRATIONS
(1.12)
X +
lv 2?
— x + 2x =
trr* L
in which v is the kinematic viscosity.
(C) A SYSTEM WITH A MORE COMPLICATED MECHANICAL
ARRANGEMENT
The system shown in Fig. 26 consists of two springs and a damper
attached to a step pulley that rolls withoutslipping on a rough surface.
I,M
/WVVV
Figure 26
It is a system of no practical utility and is conjured up merely to illustrate
our method of analysis.
The motion of the pulley can be described by the displacement of the
center mass, x, and the angular displacement, 0. Because it rolls without
slipping
Rdd = dx Rd = x + constant
in which the integration constant is zero if both 6 and x are taken to be
zero at the equilibrium configuration of the system. The system has
therefore only a single degree of freedom, and either 6 or x can be chosen
as the coordinate variable of the system.
Let M be the mass of the pulley, /, its moment of inertia about the
center of mass, and F, the frictional force at the contact. The two
momentum equations, linear and angular, are
Mx = k x x k 2 (x + rd)  C(x  rd)  F
16 = FR + C(x  rO)r  k 2 (x + rO)r
Upon substituting x by Rd and eliminating Ffrom the foregoing equations,
we have
(MR 2 + 1)6 + C(R  rfd + [k ± R 2 + k 2 (R + rf]d =
(1.13) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 55
or, if we prefer,
(MR 2 + I)x + C(R  rfx + [^R 2 + k 2 {R + r) 2 ]x =
They are in the standard form of
mx + ex + kx =
Let us now analyze the problem by energy considerations, which leads
to the observation that
Ut + V) + P =
in which P = power dissipated in damping. For this system
T = \Mx 2 + \I0 2 = \{MR 2 + I)6 2
V = ik ± x 2 + ik 2 (x + rQf
= \[k x R 2 + A: 2 (/J + rf]d 2
P = C(R rfd 2
Hence
(M/? 2 + 1)68 + C(,R  rfd 2 + [^/? 2 + A: 2 (/2 + r) 2 ]06 =
Since d ^k 0, it can be canceled out and the same equation is obtained.
1.13 Linearization of Systems in Small Oscillations
Very few physical systems are strictly linear. On the other hand, a
large number of them can be so considered if they are in motions represent
ing only small changes from their equilibrium configurations. A general
discussion on this subject is taken up in Art. 3.2. At present we shall
study a few examples.
(a) simple and compound gravity pendulums
It is well known that the motion of a simple pendulum (Fig. 21a) is
described by the equation
LB + g sin 6 =
in which L is the length of the pendulum, is the angular displacement of
the pendulum from the vertical, and g is the gravitational acceleration.
Obviously, this differential equation is not linear. Its solution is in the
form of an elliptical integral. On the other hand, if the swing of the
pendulum is limited to small angles, then sin 6=6, and the equation may
be approximated by
+ fl =
56 THEORY OF MECHANICAL VIBRATIONS
Hence the period T of the pendulum is approximately
T = —
CO
(1.13)
iWLjg
The period of a compound pendulum (Fig. 21b), having a mass M, a.
radius of gyration about its center of mass r, and a point of suspension at a
distance d from its center of mass, can be found most conveniently by
Rayleigh's method if its angular displacement is small. Let 8 m be the
amplitude of the swing, which is the maximum angular displacement, and
6 m be the maximum angular velocity, which takes place at = 0. The
Simple
pendulum
(a)
Compound
pendulum
Figure 27
(b)
maximum kinetic energy and the maximum potential energy of the system
are then
r mas = \M{r 2 + d*)dj
^max = Mgd{\  COS 6 m )
For small m , 1 — cos B m = \0 m 2 ; the system is approximately linear and
its motion, approximately simple harmonic. Hence
6 2 = a> 2 2
Upon equating the two energy expressions, we have
and
T=2n
r 2 + d 2
gd
r 2 + d 2
(b) a springloaded pendulum
In Fig. 28 a tension spring is attached to an otherwise simple pendulum
of length la. One end of the spring is fastened to a point at a distance a
above the suspension point of the pendulum. The other end is attached to
(1.13) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 57
the midpoint of the pendulum. The spring constant is k and its natural
length is a. Let us assume that the pendulum is heavy and the spring is
soft so that the equilibrium position of the pendulum is the vertical. In
terms of angular displacement 6, the energy expressions are
T = im(2adf = \(<\ma 2 d 2 )
V = 2mga(\  cos 0) + £(AL) 2 + C
Figure 28
in which AL is the change in the length of the spring from its natural
length and C is a constant that will make V(0) = 0. 48
AL = V2a 2 + 2a 2 cos 6  a = ail cos   l)
At = 0, AL = a, hence C = —ka 2 \2. By substitution and simpli
fication, we have, finally,
V = 2mga{\ — cos 0) — 2ka 2 cos I 1 — cos 1
Now, for small oscillations
1 — COS = — cos 
2 2
e e 2
1 — cos  = —
2 8
V = \{2mga  ika 2 )0 2
48 Potential energy expressions are determined only to within an additive constant.
In order to use Rayleigh's method, this constant must be such that V vanishes at
equilibrium configuration.
58 THEORY OF MECHANICAL VIBRATIONS (1.13)
Rayleigh's method then yields
2mga — ka 2 jl j g k
4ma 2 v 2a 8m
It is interesting to note that the spring in this system exerts an influence,
as if it had a negative spring constant, because its potential energy is a
maximum instead of a minimum at 6 = 0. In order for the system to be
stable at = 0, it is necessary that
ka 2 "
2mga > —
Otherwise co becomes imaginary, and no oscillation around 6 = is
possible. As a matter of fact, 6 = then would not be a stable equilibrium
configuration. On the other hand, as long as this inequality holds, it is
possible to obtain a very "soft" system with a very low natural frequency.
This scheme of having two restoring forces working against each other to
obtain the effect of a soft spring has practical applications, since a single
soft spring is usually inconvenient to incorporate in a mechanism.
(C) A BETTER APPROXIMATION
The foregoing examples belong to a class of nonlinear systems whose
differential equation of motion is of the type
mx + K(x) =
where K(x) is an odd function that may be represented by
K(x) = kx + k'x 3 + R
The remainder R is of the order x 5 . The corresponding expression for
potential energy is then
V = K(x) dx =  kx 2 +  x* + \R dx
As a first approximation, let us take
K(x) = kx and V(x) = \kx 2
The frequency w is then Vk/m. To get a better approximation, we can
take
K(x) = kx + k'x 3
and
V(x) = \kx* + \k'x'
(1.14) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
The first integral of this secondorder differential equation is
V m = T+V T=V m V
59
or
mx* = k(xj  &) + k'(xj  a*) = k(xj  x>)
From this we have
dt =
'4<
*J + * 2 )]
dx
Vk(xJ  x 2 )/m I ,{
dx
Vk(x m *  v*)\m
[ X +2k {X  +x2)
[l £*? + *
If this equation is integrated between the limits x = and x = x m , the
lefthand side becomes the quarter period
T = jT___k^_ p» (xj +
4 2co 4co& Jo Vx 2 
a; 2 )
— 1 x,
2co\ %k '
Vx^ — X'
)
<&;
in which co = Vk/m is the frequency in the first approximation. The
relation
2n t
T=—{\
CO
3k'
8A;
can be considered as a second approximation. To check its accuracy, we
take the case of a simple pendulum swinging a total angle of 60°.
T= V(Z/i)2^1 + ^) = 6.39lVZfe
r 2
The exact answer is
T
= W^J"'
^
VI  sin 2 15° sin 2
For a 120° swing the formula is in error by 3 per cent.
= 6.392VL/g
1.14 Piece wiseLinear Systems
The type of system to be discussed in this article has vibratory motions
that are governed by several linear differential equations, each of which is
60 THEORY OF MECHANICAL VIBRATIONS (1.14)
applicable only for a certain range of values of displacement or of velocity.
Strictly speaking, such a system is not linear, since it has few of the
important properties of linear systems. Its analysis is usually difficult. A
few simple cases are introduced here to give us some feeling of such
systems.
(a) a system with an unsymmetrical restoring force
In Fig. 29 a mass m vibrates between two compression springs k ± and fc 2 .
The natural lengths of the springs are such that only one of the springs
'Mwd
i\/vv\/vwwvf^~ " Vwvvwwvww
Figure 29
is under stress at any given time. The differential equations governing the
motion of m are then
mx + kjX = for x >
and
mx + k 2 x = for x <
To solve these equations, let us assume x = and x = i > at
t = 0. For the time being, we shall look only for a solution that is valid
for a certain time interval containing the instant t = 0. It is not difficult to
see that
x(t) = Q sin cojt for < t < t ±
x(t) = C 2 sin co 2 t for > t > t 2
could be made to satisfy the equations of motion and the initial condition,
at least for the certain time interval t 2 < t < t v Since x Q is positive,
co ± = Vkjm and a> 2 = Vk.Jm. Since velocity must be a continuous
function of t, at t = 0,
Xq — COiCi — CO oL 2
or
r> _ ^° r — ' T °
c x — — c 2 —
The limits f x and r 2 are determined by the condition that x{t) should not
change its sign in < t < t l9 nor in > t > t 2 . Hence
7T 77
CO, CO «
(1.14) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
The solution is therefore
x(t) = — sin corf for < t <
Oh
x(t) = — sin aj 2 t for < t <
Obviously, the motion is periodic with period
T=t 1 t 2 = 7r( +)
\co 1 0) 2 /
61
Figure 30
The solution outside the fundamental interval can be expressed as
x{t) = — sin co^t  nT)
if there is an integer n satisfying
< co^t  nT) < 77
Otherwise
x
x(t) = — sin co 2 (/ — nT)
a> 2
in which n is an integer satisfying
— 7T < co 2 (t — nT) <
A graphical representation of the solution is shown in Fig. 30.
Although the solution for free vibrations of this nonlinear system is
comparatively simple, that of forced vibration is algebraically complicated.
62 THEORY OF MECHANICAL VIBRATION 0.14)
Suppose the system is subjected to a harmonic force F cos (oj f t — p) and
its initial condition is x(0) = 0, x(0) = x > 0. The differential equation
valid for a certain time interval < t < t t is
mx + kjX = F cos (co/ — fi)
The solution of this equation can be written as
x(t) = Q cos (oj ± t — a x ) + A x cos (a> f t — ft)
where
v
ma) f 2 + Atj
Q cos a x + A x cos P =
and
i n — /l, o) f sin tf
tan a x = / /
— /t^ COS p
The upper limit t x of the time interval for which the solution is valid is the
smallest positive t x that satisfies
x(t^) = C x cos (co^ — a 2 ) + A x cos (co f t — /3) =
and
xitj <
In conjunction with the condition x(0) = we have
cos a x cos /?
cos (w^! — a x ) cos (cofa — P)
=
Solve for the smallest positive root of t x in this equation and obtain
x x — x(tj). Use this as our new initial condition and repeat the process for
mx + k 2 x = F cos [co f (t — r x ) — f3]
with
x(t^) = x{t^) = x x t x < t < t 2
This example shows that for a nonlinear system we cannot always speak
of transients or steady states, even when this system is capable of free
periodic motion and the applied force is a periodic force.
(b) a springmass system with coulomb damping
In Coulomb damping the damping force is constant in magnitude but
always in a direction that resists motion. When there is no motion, its
magnitude may assume any lower value that is enough to resist impending
motion. Such damping force is an idealization of forces due to dry friction.
(1.14)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
63
Suppose we have a simple springmass system subject to a Coulomb
damping force of magnitude/, as shown in Fig. 31. Let x be the displace
ment of mass m from its equilibrium position in the absence of/. The
vibration of this system is then governed by the following set of equations:
mx + kx = — /
mx + kx — f
for x >
for x <
Or more concisely,
mx +/sgn (x) + kx =
The symbol "sgn" stands for "sign of." It denotes a function that takes
yWv>Aj
V///////////M
x>0
±r
/ i
/\/\/\/\A
Figure 31
the value +1 or — 1, according to whether the argument is positive or
negative. 49 Hence
sgn(0) = 0/0
It is also customary to define sgn (0) = 0, although in our case the
friction force is undefined when x = 0. The last way of writing the
equation reveals more clearly the nonlinear nature of the system, but it
is more convenient to deal with our original way of writing two equations
in solving the problem.
Without the loss of generality, let us assume that the system is set into
49 The second term of (83) is such a function. Also, the differential equation of motion
for the system studied in (a) of this article can be written as
mx +
(k x + k t k x  k 2
s g
n (x)\x =
64 THEORY OF MECHANICAL VIBRATION (1.14)
vibration by giving the mass a positive displacement D and releasing it.
The initial condition is then
x(0) = D and x(0) =
Let us assume further that
f
k
The restoring force kD is thus greater than the friction force/, and in the
initial stage the motion will have a negative velocity. The governing
equation during a certain time interval < t < t x becomes
mx + kx =f
x f oj 2 x = co 2 d
Its solution under the given initial condition can be seen as
x = {D — d) cos cot + d for < t < t Y
The vibration is therefore a simple harmonic motion about the point
x = d, with amplitude D — d, and circular frequency to. This solution
is valid only as long as
x = — oj(D — d) sin cot <
Hence r x = tt\ cq, which is the half period of the free vibration. At t = t lt
x x = 2d — D. If \x x \ > d or D > 3d, the restoring force kx 1 is again
greater than the friction force /, and the mass will eventually acquire a
positive velocity. The subsequent motion is governed by
mx + kx = —f
x + co 2 x = —co 2 d
together with the condition
xitj) = D  2d and ac('i) =
For the time interval t x < t < t 2 the solution becomes
a; = (D — 3d) cos cot — d
in which t 2 = 27r/co.
Repeating this procedure, we perceive that the motion can be described
by patching together simple harmonic motions all having the same
frequency co, which is the natural frequency of the springmass system.
Each of these motions lasts only for half a cycle during which the velocity
does not change sign. After each half cycle the amplitude decreases by an
amount equal to 2d = 2f/k, and the midpoint of the cycle shifts from
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
65
(1.14)
x — d to x = — d, or vice versa. The motion stops at the end of a half
cycle whose amplitude is less than 2d. A graphical representation of the
motion is shown in Fig. 32.
The forced vibration of a system with Coulomb damping is difficult to
analyze. An exact analysis by J. P. Den Hartog is too elaborate to be
given here. We present instead the following approximate analysis. First,
let us observe that if the Coulomb damping force is only of moderate
(motion
stops at 8)
Figure 32
magnitude a harmonic force acting on the system will produce, approxi
mately, a harmonic motion. In other words, we expect motions due to
spurious disturbances to die out quickly through the various dissipation
mechanisms inherent in any real physical system, whereas a more or less
periodic motion will persist through the action of the harmonic force
applied. For a small Coulomb damping force this periodic motion can be
approximated by a sinusoidal motion. Let the resulting motion be
represented by
x(t) = A cos (a> f t — a)
The energy dissipation through Coulomb damping per cycle is evidently
4/^. On the other hand, the energy dissipation per cycle for viscous
damping is Trc^ 2 ^, according to (57). Hence we may obtain the equi
valent viscous damping coefficient by
Af\X\ = *c\X\*<o,
or
¥
C =
\0) i
66 THEORY OF MECHANICAL VIBRATION (1.14)
Substituting this equivalent damping factor into (41) or (47), we have
A =
Fo
Solving this equation for Ul, we have
or
w
Jk
~ (*)'
—moj f 2 + k 2
w.
>
■ ( l Y
8, t
(o>,Y
In examining this equation we find that the damping parameter is now the
ratio between the friction force and the amplitude of the applied force
f/F . When this ratio reaches the value 77/4, the equation can no longer be
used, since it yields imaginary values. Within the range of f/F values for
which the equation can be used the amplitude tends to infinity when the
frequency of the applied force approaches the natural frequency of the
system. The existence of a limiting value of f/F for the validity of the
analysis and the occurrence of an infinitely large amplitude in a system
with energy dissipation mechanism can be explained as follows. According
to (56), the work done per cycle by the external force is nF \A\ sin a and the
work dissipated by friction force is 4/UI. Since both are proportional to
UL the system can achieve an energy balance only by adapting its motion
to a proper phase lag a. 50
sin a = — ^
ttF
If the friction / is so large that sin a has to be greater than unity, a
sinusoidal response is completely out of the question, even as an approxi
mation. Experiments and exact analysis show that for large values of/the
response motion has "stops"; that is, during a cycle the mass will come
50 Here we are accepting a metaphysical principle, which is borne out in a number of
physical laws. This principle in essence says that nature is clever but lazy. Within the
freedom allowed her, she always finds the way in which she can exert herself the least
in reacting to changes imposed on her.
(1.14)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
67
to a complete stop or stops before moving further. Such motions are
qualitatively described in Fig. 33. The phase angle a defined by
a = sin
77Fn
is multiplevalued. Let the principal values of the foregoing equation be
a x and a 2 with
a 2 > a x
and
ai 4 occ
^t
^t
Analogous to the forced vibration of an undamped system, there is a
sudden jump of the phase angle from ol 1 to a 2 as the frequency of the force
passes through the natural frequency of the system. This jump discon
tinuity exists even in the exact solution, although there the phase angle
also varies with frequency ratio elsewhere. In a real physical system the
68 THEORY OF MECHANICAL VIBRATION (1.15)
value of a must change rapidly, and near resonance condition it assumes a
value between a x and a 2 , say a', Since
sin a' > sin a x = sin a 2
the energy input is larger than that which can be dissipated through
Coulomb damping, and the amplitude grows excessively.
The accuracy of this approximate analysis depends upon both the
friction ratio f/F and the frequency ratio. Generally speaking, the accur
acy is good when the amplitude is large. This makes the analysis of
practical value, since usually only the presence or the absence of large
amplitudes is of engineering interest. As a rough guide, if the analysis
yields a magnification factor \M/d st greater than 1.5, the error involved is
only a few per cent.
1.15 Theory of Galvanometer and MovingCoil Instruments
(a) differential equation of operation
Movingcoil instruments of different designs, typified by a D'Arsonval
galvanometer, are widely used for detecting and recording electrical
signals from a variety of sensing elements, such as strain gages, thermo
couples, and vibration pickups. To a high degree of approximation such
instruments behave like ideal linear vibratory systems having one degree of
freedom.
Basically a D'Arsonval galvanometer movement consists of a moving
coil of moment of inertia/, a magnetic field that exerts a torque on the coil
proportional to the current flowing in the coil, and an elastic spring that
produces a restoring torque as the coil turns (See Fig. 80). The motion of
the coil may be damped by immersing it in a viscous oil. In the more
delicate types used in recording oscillographs the restoring torque is
provided by the elasticity of the coil itself instead of a separate spring.
A damping torque is also produced electrically in the following manner.
When the coil turns with an angular velocity, it acts as the armature of an
electrical generator and thereby generates a back electromotive force
(emf for short). This emf causes a back current to flow if the galvanometer
circuit is closed, and this back current produces a torque opposite but
proportional to the angular velocity in the same way as a viscous damping
torque. Since the back current is inversely proportional to the total
resistance in the galvanometer circuit, so is the equivalent damping
constant. If the circuit is open, the total resistance in the circuit is infinite,
and the oscillation of the coil is therefore not damped electrically. [See
also Art. 3.14(b).]
(1.15) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 69
The differential equation of motion of a D'Arsonval galvanometer
without viscous damping is therefore
^R + R g R + R g U
6 = deflection angle
J = moment of inertia of the coil
k = spring constant of the meter movement
C = deflecting torque per unit current
e{t) = emf generated by the sensing device
R g = resistance of the galvanometer coil
R = resistance external to the galvanometer
B = back emf per unit deflection velocity 51
In practice, it is neither convenient nor necessary to describe the
characteristics of a galvanometer by so many constants. The following
four quantities are usually given in the specifications :
S — galvanometer constant or sensitivity which is the galvanometer
deflection per unit current when both are steady
co = natural undamped (circular) frequency
R c = critical damping resistance, that is, external resistance required to
produce critical damping
R g = resistance of the galvanometer coil
The relationships between these two sets of constants are
c c 2 k
S =  or = 
k J
c c = iVjk (ii)
Rc + *a
By dividing the original equation by k and utilizing the foregoing relation
ships, we have
co 2 co\R + Rj R + R (
e(t) (iii)
Having written the equation in this form, we realize that the units of
measurement of 6 can be arbitrary as long as the same unit is used in
defining S. For a given setup of instrumentation we can also lump
together S/(R + R g ) as the deflection per unit signal strength and consider
e as the signal strength.
51 The constants B and C are equal. This fact is shown in Art. 3.14, but it is of no
consequence here.
70 THEORY OF MECHANICAL VIBRATION (115)
(b) FREQUENCY RESPONSE AND OPTIMUM DAMPING
In an ideal instrument the response d(t) should be proportional to the
signal e{t). In an examination of the differential equation (iii) it is seen
that such idealized characteristics can be approached if oj is high enough
so that the first two terms in the equation are small. This means that the
undamped oscillation of the galvanometer must be relatively fast in
comparison with the rate of signal change, or
\e\ > \e/(o\ \e\ > e?/w 2 
However, in practice, this desirable condition is not easy to achieve
without sacrificing the sensitivity of the meter. This is because a high
value of co implies a stiff spring or a light coil ; a lighter coil means fewer
turns, smaller coil loop, or finer wire, and finer wire results in higher R g .
Any of these conditions result in less deflection per unit signal strength. 52
Therefore, in choosing galvanometers for instrumentation a compromise
has to be made between sensitivity and fidelity of response.
Although the inertia and elastic stiffness of the galvanometer movement
cannot be arbitrarily varied because of constructional practicability,
the amount of damping is relatively easy to adjust; and, by a proper
adjustment of damping force, the performance of the instrument can be
much enhanced. First, suppose the galvanometer is to record steadystate
sinusoidal signals of different frequencies. Evidently, it is desirable for
the amplitude of the response to be proportional to that of the signal,
regardless of its frequency. In other words, in the ideal situation the
magnification factor should be independent of the frequency ratio. Figure
16, however, shows that this is not exactly realizable in a simple spring
mass damper system. The most we can accomplish is an approximately
constant amplitude of response for a limited frequency range. Such a
condition exists for instruments with a moderate amount of damping at
signal frequencies below the natural undamped frequency of the instru
ments. Figure 34 shows the magnified lower left portions of the curves
in Fig. \6a. It is seen that some of the curves are reasonably "flat" within
the frequency range shown. We noticed also that for high values of cjc c
the point (0, 1) is a maximum and for low values of cjc c it is a minimum.
As shown in Exercise 1.17, the transition takes place at cjc c = l/\ 7 2 when
the point (0, 1) becomes a minimax, so that both the slope and the curva
ture are zero, and the curve has a broad plateau. This value of damping is
52 A way of compensating the loss sensitivity in using a lighter coil and stiffer suspen
sion is to increase the magneticfield strength of the pole pieces. This usually results in
bulkier instruments. Compactness is of importance in such instruments as oscillo
graphs, which contain scores of galvanometers in a small enclosure.
(1.15)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
71
then a theoretical optimum. In practice, a lower damping ratio is usually
chosen because, by allowing the curve first to go up a little and then to
come down, the usable frequency range is further extended, although it is
not so flat at the beginning as the curve corresponding to the theoretical
optimum damping of 0.707. For instance, with 65 per cent of critical
1.016
1.012
1.008
1.004
1.000
0.996
0.992
0.984
c/
V (
).60v
C K
= 0.65\
\
N c /
c r = (
).70_
\
\
Sc/c c
= 0.75 N
\
c/c c  0.80\
1 1 \
\
0.10 0.20 0.30 0.40 0.50
Frequency ratio ooJcc
Figure 34
0.60
0.70
damping, the variation in frequency response for signal frequencies up to
60 per cent of the natural frequency of the instrument is within ±1.3 per
cent.
It must be emphasized that even within the frequency range, where the
frequency response is relatively constant, the phase difference between the
signal and the response varies appreciably. Hence, if the signal contains
harmonics of different frequencies, their phase differences will not be
preserved in the response. This results in a distortion of the waveform
called a. phase shift. This type of distortion may or may not be important
in a given application of the instrument.
The criterion for optimum damping can also be based upon the response
of the instrument to step signals. This is especially appropriate if the
instrument is to record transient signals faithfully. Let S be the sensitivity
of the instrument. The response of an ideal instrument to a step function
S(t) is then SS(t), or simply S for t > 0. 53
53 We assume here that the signal is in the form of a current. If it should be a step
voltage, replace S by SI(R + R„). In any event, by sensitivity or meter constant one
means the response per unit signal strength when both are constant with respect to time.
72 THEORY OF MECHANICAL VIBRATION (115)
The response of the actual instrument is u(t), which is given by (696)
with \\k replaced by S. We can then define the error E{t) as
m . ,  f
It can be verified that
E{t) =
— £?< tan W^ cos (gU  (5)1
0) r . J
The time average of the error in a given time interval r is
\[ mdt
and the mean square error is
i f T {£(r)} a A
T JO
Other things being equal, these integrals are a function of c/c c ratio. The
criterion for optimum damping may then be the minimization of these
integrals in a specific interval.
With a moderate amount of damping the error function E(t) becomes
quite small after t reaches a value equal to a few times the natural
period of the instrument, so it is quite reasonable to optimize the amount
of damping by minimizing either of the following two integrals:
I* oo C oc
\E\dt and E 2 dt
Jo Jo
Both of these improper integrals are convergent.
The first integral cannot be evaluated conveniently. 54 Since this cri
terion is somewhat arbitrary and since the results, if carried out, cannot
differ much from those by minimizing the second integral, we choose the
second criterion instead. This leads to an optimum damping ratio of
cjc c = \. The details of this procedure are left to the readers as an exercise.
Another and perhaps commoner procedure of choosing an appropriate
damping ratio is by minimizing the socalled response time, that is, the
54 In case one wishes to evaluate it, the integral can be expressed as an infinite series
of definite integrals, each covering an interval of time in which the cosine function does
not change sign. Except for the first term, the values of the succeeding integrals form a
geometrical progression with a common ratio of eA/ 2 . The sum of the series can thus be
computed.
(1.16) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 73
time required for the instrument to settle down into an oscillation between
prescribed limits. This procedure usually amounts to the determination
of the minimum damping ratio, so that the maximum "overshoot" is less
than the amount prescribed, although the two criteria are based on two
slightly different physical considerations. 55 For example, with a prescribed
error of ±5 per cent, the shortest response time is 43 per cent of the period
of natural oscillation, and it corresponds to a damping ratio of 0.69, which
is also the value of the minimum damping ratio for a maximum overshoot
of 5 per cent. This overshoot takes place at t — 0.65T.
The different considerations involved in selecting an optimum damping
ratio, as discussed, lead to the observation that for a general purpose
instrument a damping ratio of 0.65 is a good compromise.
In concluding this article, let us remark that although a moving coil
galvanometer was chosen for this study many of the results, observations,
and methods of analysis are applicable to other types of instruments and
control equipment.
1.16 Seismic Instruments and Transducers
The type of device discussed in this article is essentially a singledegree
freedom vibratory system mounted on a frame or in an enclosure which
can be attached to a moving body whose motion is to be measured or
converted into an output signal. The total mass of the system and its
mounting must be relatively small so that it will not affect the motion of
the body to which it is attached.
(a) seismograph
The oldest instrument of this type is the seismograph, an instrument
for detecting the motion of the earth's surface. In its simplest form it
consists of a springmass system of very low natural frequency enclosed
in a frame attached to the ground, as schematically shown in Fig. 35. 56
The input to the system is the motion of the ground (hence of the frame),
and the output is the relative displacement between the frame and the mass.
This relative displacement can be recorded by attaching a tracing pen to
the mass and attaching graph paper to the frame.
Let x be the displacement of m, x s , the displacement of the frame, and
68 See Draper, Mckay, and Lees, Instrument Engineering, Vol. II, McGrawHill,
New York, 1953, pp. 264265.
56 We assume in Fig. 35 that the motions are in the vertical direction. The mass
could be suspended to have freedom in three dimensions. However, since the three
motions are independent of one another, the principles involved are essentially those of
a singledegreefreedom system.
74
THEORY OF MECHANICAL VIBRATION
(1.16j
x r , the relative displacement x s — x. The differential equation of motion is
then
mx + k(x — x s ) =
or
mx r + kx r = mx s
x r + oj 2 x r == £ s
From this equation it can be inferred that if the natural frequency of the
system is very low, relative to the two accelerations, then
x\ == x\
Ground
Figure 35
This implies that the response is equal to the signal when everything starts
from a rest condition at t = 0. To achieve this approximate relationship
physically, we need to have a very heavy mass 57 suspended by very soft
springs. As the frame moves because of earthquake or underground
explosion, the heavy mass stands almost still, and any rapid vibration of
the ground is reflected directly by the relative motion between the mass and
the frame. The principle of a seismograph is similar to that of certain
vibration pickups.
(b) vibration pickups
There are several types of vibration pickups whose construction is
essentially a springmass system (with or without a damper), mounted in
a housing which can be attached to a vibrating machine or structure, as
shown in Fig. 36. We give them the generic name seismic instruments
because they are related to seismographs in their operational principle.
57 Even a very heavy mass is still not heavy enough to affect the motion of the earth's
surface.
(1.16)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
75
The characteristic feature of a seismic instrument is that its output or
response is generated by the relative motion between its housing and the
mass inside. This output may be in the form of a mechanical movement, a
tr 1 ^
*~x s
Vibrating structure
Figure 36
N
AA/W!
— . i
Coil
o *■
8 Output
o *
\
/
Moving coil pickup
Output
v/W
Crystal
pickup
Piezoelectric
.crystal or other
stressorstrain
sensitive device
Variable reluctance pickup
Insulator
Large
resistance
Variable
capacitance
pickup
Figure 37
moving light beam, or an electrical voltage. Figure 37 shows schematically
some typical instruments whose output is in electrical form.
76 THEORY OF MECHANICAL VIBRATION (116)
The strength of the output may depend upon the relative displacement
or the relative velocity between the housing and the mass, and accordingly
the pickup is said to be displacementsensitive or velocitysensitive. The
response may be intended as a measurement of the displacement, the
velocity, or the acceleration in the motion being measured, and accordingly
the instrument is called a vibrometer, a velocity pickup, or an accelerometer.
Regardless of what we consider as the input signal and what the pickup
gives as response, the basic differential equation for a seismic instrument is
mx + c(x — x s ) + k(x — x s ) =
in which x is the displacement of m and x s is the displacement of the
motion to be sensed by the pickup. Let
and we have
mx r f cx r f kx r = mx s (i)
If the instrument is a vibrometer and it is displacementsensitive, as the
seismograph previously described, (i) is the relationship between the
signal and response, and the frequency response of the vibrometer is
described by
Tim,
w (i .
\V Vib
1 
mojf + k + icco f
k + icco f
k — mco f 2 + icto f
= 1  T r (ko f ) (ii)
in which A s and X r are the complex amplitudes of the signal and the res
ponse, respectively.
According to (80), the function T r is the transfer function for the trans
missibility of the vibratory system inside the seismic instrument. With
the help of Fig. 22, we may conclude the following:
(1) Since T r approaches zero as co f becomes large, the response is
nearly constant if the natural frequency of the instrument is sufficiently
below that of the vibration being picked up. On the other hand, the
instrument is insensitive to slow changes in the signal. A signal such as
the one shown in Fig. 38a is picked up by the instrument as one represented
by Fig. 386. This can be an advantage or a disadvantage, depending upon
applications. Note that the situation here is exactly the opposite of
galvanometer response.
(2) Since T r approaches zero faster with less damping, there is no
(1.16)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
77
advantage in introducing any damping in the system. Therefore, a
vibrometer is designed with a minimum amount of damping.
Let us now take the case of a velocity pickup whose output is propor
tional to the relative velocity of the motion inside. The signal is then taken
to be v s = x s and the response to be v r = x r . Evidently, by differentiating
(i) with respect to /, we have
mv r + cv r + kv r = mv s
So if this time we let our signal k 8 e la>ft be v s and the response X r e l<0 f l ,
MHA^lt^
(b)
Figure 38
v r , the relation between X r and X s would be the same as before, and all the
observations we made about a vibrometer would hold true for a velocity
sensitive velocity pickup.
Now let us suppose that the instrument is an accelerometer and its
output is proportional to the relative displacement. The input signal is
then
and the response is
l s e % ^ 1 = x s
;./<v = Xr
According to (i), the relationship between X r and X s is then
( — mcQ f 2 + ico) f + k)X r = mX s
T a (ia)
<>©
m
k — mco f 2 + icoo f
co f 2 2c co f
1 ! + * —
or c c oj
(iii)
78 THEORY OF MECHANICAL VIBRATION (1.16)
The denominator in this expression is the same as that of (46), and the
numerator is a constant of the instrument. The response of an accelerom
eter is therefore proportional to its magnification factor, and the
situation is the same as a D'Arsonval galvanometer. To repeat what we
said about a galvanometer:
(1) A displacement sensitive accelerometer should have a natural
frequency that is high in comparison with that of the signal. On the other
hand, by making its natural frequency high, one sacrifices its sensitivity,
which, according to (iii), is proportional to 1/w 2 .
(2) For best results an accelerometer has to be damped to a c\c c value
between 0.6 and 0.7. The damping in an accelerometer serves two pur
poses. The first is to extend the usable frequency range of an instrument
of a given sensitivity. The second is to prevent resonance with the higher
harmonics of the input signal. For instance, if the accelerometer has a
natural frequency of 120 cycles per second and the signal is a slightly
distorted sinusoidal motion of 40 cycles per second, the instrument
without damping would be resonant with the third harmonic of the signal
and give entirely erroneous readings. Note that in a vibrometer the same
situation will not be obtained if the natural frequency of the instrument is
below that of the fundamental frequency of the signal.
Another remark we made about the phase shift in a galvanometer with
damping also applies here; that is, by extending the frequency range of
the instrument through proper damping, we do not eliminate the phase
shift distortion, which is important only when the waveform of the signal
is of interest.
(C) COMPENSATION AND EQUILIZATION
Most modern instruments give their outputs in the form of electrical
signals. These have to be converted by external means into the motions
of a pointer or recording pen. The conversion device is in itself a signal
transferring system. It is then possible to introduce an electrical network
into this device, which has a signaltransfer characteristic compensating in
part for the inaccuracies in the sensing instrument. For instance, if an
accelerometer is damped to 70 per cent of the critical, its response falls off
for signal frequencies above 20 per cent of its natural frequency. By
introducing a network whose response shows a proper rise above the same
frequency, we can extend the usable frequency range of the combination.
Another familiar example is a magnetic phonograph pickup, which is
essentially a velocitysensitive seismic instrument in which the arm is
equivalent to a seismic mass and the needle is equivalent to a mounting
frame. Since most records are cut to be played by displacement sensitive
(1.17)
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
79
pickups, the use of magnetic pickups requires compensation in the play
back amplifier.
The point to be emphasized is that with the available flexibility in
compensating network design a flat frequency response of a sensing device
may not always be of the utmost importance as long as its response
characteristics are known.
1.17 Vehicle Suspension
The suspension system of a vehicle may be analyzed in the first approxi
mation as a vibratory system having a single degree of freedom. As shown
*rk
m
1 ^
k
> y = vt
Figure 39
iii Fig. 39, the system is equivalent to a seismic instrument, and its differ
ential equation of motion is
or
mx r + c(x r — x s ) + k(x r — x s ) =
mx r + cx r f kx r = cx s + kx s
(0
(ii)
where x s and x r are the vertical displacements of the wheel and the vehicle,
respectively. If the profile of the road surface is given by x s =f(y) and the
velocity of the vehicle is v, then
x s =f(vt) and x s = vf'(vt) (iii)
So the lefthand side of (ii) is a function of /. When the road surface
irregularity is more or less periodic, the frequency response of the system
is of immediate interest. It must be pointed out that although the differen
tial equation is the same as that for a seismic instrument the response of
80
THEORY OF MECHANICAL VIBRATION
(1.17)
the system is taken to be the motion of the vehicle body rather than the
relative displacement. The frequency response is then
() =
k + icoOf
k — mo) f 2 + ico).
= T r (i(o f )
which is the transmissibility of a springmassdamper system. The
frequency co f is determined by the profile of pavement irregularities as
well as the speed of the vehicle v, A look at Fig. 22 enables us to make
the following observations :
1
i
x s =J~(t)
Figure 40
(1) Insofar as vibration absorption is concerned, it is desirable to use
soft suspension springs to lower the natural frequency of the system.
Very soft suspension springs, of course, have problems of their own.
These include lateral instability and excessive changes in road clearance
and vehicle height between loaded and unloaded conditions.
(2) At low speeds, which correspond to frequency ratios below 1/V2,
damping is necessary to cut down the vibration transmitted from the wheel
to the body, whereas at higher speeds the existence of damping is actually
disadvantageous insofar as steadystate road disturbances are concerned.
An important consideration in the design of vehicle suspension concerns
its ability to absorb bumps and holes in the pavement. This ability can
best be assessed by its indicial response. By definition, the indicial response
in this case is the solution to (ii) for x s (t) — S(t) with x r (0) = 0. There is,
(1.17) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 81
however, a mathematical difficulty in solving this problem by the classical
method, since with x s being a step function cx s is infinite at t = 0, although
it is zero at all other times. There are other methods, including the use of
(96), whereby this difficulty can be removed. But we have dealt with this
problem before in Art. 1.11, in which it is shown that the differential
equation (78) is equivalent to a pair of equations (77). Borrowing the
subsequent results, together with {69b), we have
u(t) = ci— e  {cl2m)t sin co c t) + ( 1  — e~ {cl2m)t cos (co c t  6))
\mco c I \ co c /
With the help of (27), this expression can be simplified to yield
(iv)
u r {i) =
1
CO
co c
e (c/2m)t
cos
(co c t + 6)
From this
we
have
u r \t)
=
CO 2
— e
co c
(c/2m)« ^
I (co
J + 26)
(v)
Now we see that at t = the foregoing expression does not vanish, since
,,^ 0j2 • ~ c 2coc c
u r '(0) = — sin 2d = = 
co c c c m
Hence this does not satisfy the initial condition x r (0) = 0. However,
when we examine the physical meaning of indicial response, we see that
the system is assumed to have zero displacement and zero velocity for all
t < 0. They are zero at / = only if they are both continuous at t = 0.
Such continuity exists if the signalresponse relationship is of the form (97);
but, when the relationship is of the form (98), the derivatives of the
indicial response may not be continuous. Returning to the problem on
hand, we see that when the wheel is suddenly given a displacement the
body of the vehicle will suddenly acquire a velocity of c/m. This statement
is true also for c\c c > 1 , since (iv) and (v) are true if co c and 6 are allowed to
be complex.
This sudden velocity change 58 of the vehicle body represents a shock or
impact which should be kept small. This means that the value for c/c c
should not be too large. On the other hand, to prevent large oscillation,
c\c c should not be too small. A good compromise is for c\c c = 1.
68 In reality, the tire and wheel structure is not absolutely rigid so that the acceleration
will not be infinite.
82 THEORY OF MECHANICAL VIBRATION
1.18 Structural Damping and the Concept of Complex Stiffness
(1.18)
In a real vibratory system the restoring force of the "spring" is not
entirely conservative. Thst is to say, cyclic deformation of the spring
consumes mechanical energy. The dissipative forces involved originate
either in the interior of the spring material or at regions between two pans
of a builtup spring. The outward manifestation of such dissipative forces
is described by the general term structural damping. As a rule, forces due
4 Stress
Strain
Figure 41
to structural damping are small, but often their presence affects the dynamic
behavior of a vibratory system, especially if the vibration is selfexcited. 59
In this article we discuss a simple form of structural damping and its
effects on the vibratory motion of a singledegreefreedom system.
It is generally accepted that when structural damping is caused by the
material of the spring in a springmass system the hysteresis of the spring
material under cyclic deformation is responsible for the energy dissipation.
Hence we have the term hysteresis or hysteretical damping. Precise measure
ments on the stressstrain relationship of most real materials show that
even at stress levels much below their accepted elastic limits cyclic straining
produces hysteresis loops such as that shown in Fig. 41 . The area enclosed
by the loop represents the mechanical energy dissipated by the material
during one complete stressstrain cycle.
59 A vibration is said to be selfexcited when the external force becomes a periodic
excitation by virtue of the vibration itself. For instance, the stroking of a bow on a
string produces a periodic force on the string as the latter vibrates. Also, the flutter of
an airfoil is excited by aerodynamic forces which become periodic as the airfoil flutters.
(1.18) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 83
(A) EXPERIMENTAL EVIDENCE
Using a rotating beam loaded by a dead weight, Kimball and Lovell 60
showed that for most materials the energy dissipated per cycle is indepen
dent of frequency over a wide frequency range and proportional to the
square of the stress (or strain) amplitude of the cycle. This observation
can be translated to mean that the hysteresis loop is not affected by the
rate of straining and that a change in the stress amplitude produces a
change in the size of the loop but not its shape. More modern measure
ments by Lazan 61 indicate that the foregoing statement is approximately
true for many materials in a limited range of frequencies and stress
amplitudes. The actual phenomenon, of course, is exceedingly complex
and is dependent upon a host of mechanical and metallurgical factors.
(B) STRUCTURAL DAMPING FORCE IN STEADYSTATE VIBRATION
When a springmass system with a small amount of structural damping
is excited by an external harmonic force, it is reasonable to expect that the
resulting steadystate vibration will be a harmonic oscillation with the
frequency of the excitation co f . Let the amplitude of this oscillation be
A. The law of structural damping, according to Kimball and Lovell,
then states that the energy dissipation E d per cycle is
E d oc f (i)
In the meantime, according to (57), the energy dissipation per cycle by
viscous damping under the same circumstance is
E d = 7TCCO f A 2
Therefore, a system with structural damping in steadystate vibration
may be analyzed as a viscously damped system having a damping constant
c inversely proportional to the frequency. In other words, the linear
equation
mx \ x + kx = F cos cu f t (ii)
CO f
has a steadystate harmonic solution in which the energy dissipation per
cycle is independent of the frequency co f although proportional to the
square of the amplitude. The constant h is determined by the property of
60 "Internal Friction in Solids" by A. I. Kimball and D. E. Lovell, Physical Review,
Vol. 30, December 1927.
61 "A Study with New Equipment of the Effects of Fatigue Stress on Damping
Capacity and Elasticity of Mild Steel" by B. J. Lazan, Trans. Am. Soc. Metals, Vol. 4,
1950.
84
THEORY OF MECHANICAL VIBRATION
(1.18)
£ 2.0
180
150
120
90
85 60
30
0,5
1.0 1.5 2.0
Frequency ratio co^/w
(a)
2.5
S^ ^^^^"^
/ / X*
=^y/
i i
0.5
1.0 1.5 2.0
Frequency ratio co^/w
(b)
2.5
3.0
Figure 42
(1.18) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 85
the springmaterial, and it has the dimension of k. Often one writes
h = k?]
in which r\ is a nondimensional quantity called the structural damping
coefficient and is usually less than unity. The steadystate solution for
(ii) can thus be obtained by replacing c with hjco f in (41) of Art. 1.6. The
result is
x = Ul cos (co f t — a)
h ri
tan a
—tncOf 2 + k
 o
^0 d,t
\/(mco * + kf + h 2 I a)*\* , (hi)
+ rf
(■  w
Plots of the magnification ratio \k\/6 st and the phase lag a versus the
frequency ratio co f /co for different value of r\ are shown in Fig. 42. It is
instructive to compare these plots with Fig. 16, which is designed for a
viscously damped system. An interesting feature of Figure 42a is that the
peaks of all the curves are at co f lco = 1 .
(C) COMPLEX NUMBER REPRESENTATION IN THE CASE OF STEADY
STATE VIBRATION WITH STRUCTURAL DAMPING
Since (ii) is a linear differential equation having a solution of the form
(iii), we can use a complex number representation
x = Xe^f 1 (iv)
by rewriting the differential equation of motion as
h
mx \ x + kx = F e H ' J ^ (v)
Upon differentiating (iv) and putting into (v), we obtain
x = icOfXe^f 1 = ico f x
mx + (ih + k)x = F e ia>ft
or
mx + k(\ + irj)x = F e' CO}t (vi)
Writing the equation of motion in the form of (vi) has the advantage of
86 THEORY OF MECHANICAL VIBRATION (118)
simplicity as well as the disappearance of co f from the lefthand side of the
equation. The complex quantity
K = k + ih = k{\ + irj)
is called the complex stiffness? 2 This complex quantity represents the
elastic and the structural damping forces of the spring at the same time.
Such representation is particularly advantageous in dealing with systems
having many degrees of freedom. The vectorial representation of (vi) is
shown in Fig. 43.
Figure 43
(d) vibration of structurally damped system
in the absence of external excitation
It must be emphasized that all the results presented are valid only when
the vibration is known to be a harmonic oscillation. It is obvious from (ii)
that in the absence of external excitation the homogeneous differential
equation
mx \ x + kx =
(J0 f
has no physical meaning, since it includes an undefined quantity co f .
Unfortunately, co f does not appear at the lefthand side of (vi), and this
fact led some investigators into trying to solve the problem by solving the
equation
mx + k{\ + irj)x = (vii)
We must realize that (v) can be replaced by (vi) only because we know the
solution to be of the form (iv). 63
62 As a matter choice, some authors call it the complex damping.
63 See the discussion following (49) in Art. 1.7. Also, for a physical problem meaning
fully described by (vii), see Art. 2.14(c).
(1.18) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 87
The experimental evidence presented in (a) was from a steadystate
phenomenon. We really have no adequate physical data to analyze a
damped oscillation. It is, however, reasonable to believe that if the
structural damping force is small, so that the decay of the resulting damped
oscillation is gradual, the motion may be assumed to be the same as that of
a viscously damped system with an equivalent damping constant given by
_ h
oj c
The equation to be solved is then
mx \ x + kx = (viii)
co c
where co c is the damped frequency of the vibration. This approach was
originally suggested by Collar. 64 The solution to (viii) can be obtained by
replacing c with hjo) f in (19) and (19a).
x{t) = Ce {htl2m ^ cos (oo c t  a) (ix)
The damped frequency co e is to be determined by (20)
V 4mkoj 2  h 2
«r = =
2moj c
Solving for co c , we obtain
k + Vk 2 h 2 /l + VI — >7 2
M < = J 2Tn = °V 2 (X)
The oscillatory solution (ix), of course, can be valid only for r\ < 1. An
aperiodic solution for r\ > 1 has no physical meaning because we know
nothing of the law of structural damping for such motion. It appears that
according to (x) the limiting value of co c for oscillatory motion is co/Vl
instead of zero. This is not so mathematically. There is a second solution
to (ix) which corresponds to the use of minus signs instead of the plus
signs under the radicals in (x). The branch point corresponds to a loga
rithmic decrement of 277, a decaying rate too rapid to justify the extrapo
lation of KimballLovell's law, as stated.
A more lucid way of expressing the relationship between co c and r\ is by
the use of the angle d defined in (27).
co, / 1 + cos 2d
— = cos d =
o
64 See "The Treatment of Damping Forces in Vibration Theory" by R. E. D. Bishop,
Journal of Royal Aeronautical Society, Vol. 59, No. 539, November 1955.
88 THEORY OF MECHANICAL VIBRATION (1.18)
In comparing this expression with (x), evidently
rj = sin 2d (xi)
(e) complex number representation of damped oscillation
In complex number representation the solution to (viii) is in the form of
x = le ,at
where the complex frequency a is defined in (22) as
c
a = co c + i — co — oj c (1 + i tan b)
Hence
x = iax = io) c (\ + i tan b)x
By substituting this into (viii), we obtain
mx f ih{\ + / tan b)x + kx =
or
ma; + k{\ — r\ tan b + z^)x =
Because of (xi)
1 — 77 tan b = 1 — 2 sin 2 (5 = cos 2(3 = Vl — ?y 2
We have, finally,
mx + &(Vl  ?f + />?> =
mx + ke 2ld x =
or
Z7zi; + /c^ = (xii)
Therefore, in this case the complex stiffness is the quantity
Kl = k(V\  yf + irj) = ke m
Here the structural damping coefficient ?] enters not only into the imaginary
part but also into the real part of the complex coefficient. This came about
because of our scheme of representation and not because structural
damping force has any physical effects on the conservative force of the
spring. This observation is clear, as we examine the original differential
equation (viii).
We see, therefore, that for a given system with structural damping the
complex stiffness k in the case of steadystate forced vibration is different
from that in the case of a damped oscillation k ± . 65 For small v\ the differ
ence is, of course, small. It arises out of the difference in the structural
65 For another complex number representation of the same hysteresis characteristic
see Art. 2.14(c).
(1.18) SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 89
damping laws used to analyze the two types of motion. The use of k for k 1
in (xii) leads to (vii), which gives inadmissible results. 66 Myklestad, 67 by
a different method of derivation, however, arrived at an equation for
steadystate vibrations in which k x appears instead of k.
mx + ke 2id x = F e i<a f f (xiii)
He stated that certain features of his solution agree better with experi
mental observations, but there is some question 68 about the physical
reasoning underlying his method of derivation. As a practical matter,
since the structural damping force is generally small, the numerical results
obtained from (viii) and (xiii) have no significant difference.
(f) other types of structural damping
The structural damping law of Kimball and Lovell has only limited
validity. It has the advantage of yielding a linear differential equation.
Therefore, whenever reasonable, this law is used for analysis. There are
many actual systems with structural damping whose oscillations cannot
be described by linear equations without gross error. For example, for a
certain builtup beam Pian and Hallowell 69 found that hysteresis loss
caused by slipping between its parts is approximately proportional to the
third power of deformation while it is independent of the stress amplitude.
Naturally, the analysis of such a system is more complicated. Since
structural damping force can originate from a number of different physical
phenomena, no unifying treatment of such forces is possible.
Exercises
1.1. Write the differential equations of motion for small oscillations of the
systems shown.
66 For instance, the solution to (vii) is oscillatory for all values of rj and co c increases
with ?/. The solution is given in Art. 2.14(c).
67 "The Concept of Complex Damping", N. O. Myklestad, Journal of Applied
Mechanics, Vol. 19, No. 3 (September 1952)
68 See "Concept of Complex Stiffness Applied to Problems of Oscillations with
Viscous and Hysteretic Damping" by S. Neumark, Royal Aircraft Establishment,
Report No. Aero. 2592 (V.D.C. No. 533.6.013.42) September 1957.
69 "Structual Damping in a Simple BuiltUp Beam", by T. H. H. Pian and F. C.
Hallowell, Proceedings of First U.S. National Congress of Applied Mechanics, pp.
97102, 1951. See also "Structural Damping" by T. H. H. Pian, Chapter 5 of Random
Vibration, edited by S. H. Crandall, MIT Technology Press, Cambridge, Massachusetts,
1958.
90
THEORY OF MECHANICAL VIBRATION
wmm
<
(a)
WW
O m
(b)
Exercise 1.1
id)
1.2. A torsionbar suspension system for vehicles may be approximately
represented by a weight hanging on an Lshaped round bar as shown. Find the
natural frequency of the system in terms of a, b, d, E, G, and W.
E = Young's modulus G = Modulus of rigidity
Exercise 1.2
I
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
91
1.3. To find the moment of inertia of a flywheel, a certain engineer hangs the
wheel on a nail and measures the period of swing. The wheel weighs 60 lb, the
point of suspension is 10 in. from the center of the wheel. If the period of the
swing is 1.45 sec, what is the moment of inertia of the wheel about its centroidal
axis? The wheel is assumed to be well balanced.
1.4. A springloaded cam follower follows an eccentric radius r and eccentricity
e. The mass of the follower is m, and the spring has a spring constant k and a
natural length that is longer than its maximum length in position by an amount
equal to 2e.
(a) Show that for small e\r the follower has a simple harmonic motion when
the eccentric rotates with uniform angular speed oj f .
(b) What is the maximum allowable oj f so that the follower will not leave the
cam?
Exercise 1.4
1.5. Show that
A cos (cot — a) + B cos (cot — /?) = C cos (cot — y)
where A, B, and C are three sides of a triangle with the respective opposite angles
being ±(y  p), ±(<x  y), and xr ± (a  p).
1.6. A system consists of four uniform identical rigid bars each of length a
and mass m, linked together by frictionless pin joints and a linear spring of
spring constant k and natural length a. Find the natural frequency of the system
by Rayleigh's principle. Assume that the mechanism lies on a smooth horizontal
table so that its center of mass is stationary during oscillation.
92
THEORY OF MECHANICAL VIBRATION
Exercise 1.6
1.7. Draw the phase trajectory of a simple pendulum of length L swinging a
total angle of 180°. Compare its shape with that of an ellipse having the same
major and minor axes.
1.8. A plumb bob of effective length / is hung on a nail on a vertical wall.
Directly under the nail of suspension is another nail which catches the string
when the plum bob swings to one side but not when it swings to the other side.
If the distance between the nails is //4, what is the period for small angle swing
and what is the ratio of the two maximum angles the string makes with the
vertical ?
Exercise 1.8
1.9. A piston of mass m "floats" in a cylinder of height h and crosssectional
area A ; the cylinder contains a perfect gas. Assuming that the compression and
rarefication of the gas are reversible adiabatic as the piston is set into small
upanddown oscillation, find an expression for the frequency of such oscillation.
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM
93
%
\
m
Air
1
I
h
[
1
V/////////////////////////
Exercise 1.9
1.10. A small ring slides back and forth without friction along a wire bent
to a curve in a vertical plane described by the equation
x 1 x*
2a a A
Find the period of its oscillation when x max is 0.2a.
Hint.
'dsV
T = \m
dt
= hn(\ + y' 2 )x*
V = mg(y  y m ) T + V
d(T + V\
( y  Vrr
d_ y
dx
=
For small
+ g(y'  y' z + 2y m y'y"  4y m y' 3 y"  2yy'y") = o
Neglect all terms higher than third order of x.
1.11.
(a) The solution expressed by (15) can also be written as
x{t) = e^ 2m ^(A cosh wt + B sinh wt)
in which w = V(c 2 /2m) — 4mk = ia> c . By the usual method, determine
the constants A and B in terms of x and x .
(b) Utilize the formula
cos id = cosh 6
and
sin id = i sinh 6
to show that (19) and (23) will lead to the same answer as (a).
(c) Show that, as in Fig. 7, a hyperbola may be used to relate cjc c and w/a>.
If we let cjc c = cosh € and wjco = sinh <?, what is the geometrical meaning
of e?
(d) Show that (18) can be obtained by letting co c approach to zero in (19)
or by letting s x = to and s 2 approach s ± .
94 THEORY OF MECHANICAL VIBRATION
1.12. Show that the angle 6 in Fig. 8 is the 6 in (27).
1.13. Show that
In — — = A
x(t + T)
1.14. Draw the phase trajectory of a damped singledegreefreedom system
with c\c c = 0.5 and compare its shape with Fig. 8.
1.15. A suspension galvanometer has the following periods of swing: open
circuit 3.00 sec; 2000 ohms across the terminals, 3.20 sec; 1000 ohms across the
terminals, 3.60 sec. What is the coil resistance and the damping resistance of the
galvanometer? Is this a practical way to determine the two resistances of a
galvanometer experimentally? Why?
1.16. Plot two cycles of the envelope C in (36) for B = 0.7 A.
1.17. In Fig. 15, if a circle is drawn with its center at the origin and tangent
internally to the parabola, the radius of the circle and the point of tangency
correspond to the maximum frequency response and its phase lag for a given
damping ratio. Using methods in analytical geometry, show that
(a) The radius of the tangent circle is sin 26, hence the maximum frequency
response is
Aml^st = CSC 2^
(b) The corresponding phase lag a m is
a m = tan _1 ( Vcos 26 esc <5)
(c) The frequency response curve has a peak only when
6 < or <1/V2
4 c c
1.18. Improve the accuracy of the period obtained by Rayleigh's method for
the system shown in Fig. 28, when the 6 has an amplitude of 30 : and (gja) =
(k/m).
1.19. A spherical ball of weight w and radius r rolls back and forth without
slipping in a plane motion inside a bowl whose surface is a surface of revolution
formed by the sinusoidal curve
y=r\\ cos^)
Find the Rayleigh's method its period for small oscillations.
1.20. An overdamped singledegreefreedom system is set into motion with
initial displacement x and initial velocity ^ . Show that the condition for .v(t) >
at all t < oo is that x > s 2 »V
1.21. A damped singledegreefreedom system has zero initial velocity. Show
that, other things being equal, the integral
I
x*dt
is a minimum for c = 0.5c c .
Hint. Use (15) and (17) for your analysis.
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 95
1.22. The signal from a 4arm Wheatstone strain gage bridge is to be recorded
by an oscillograph using a suspension coil galvanometer which has the following
specifications: galvanometer coil resistance 60 ohms, critical damping resistance
140 ohms, and natural frequency 120 cycles. If the gage resistance is 360 ohms
and the record shows a steadystate sinusoidal signal of 90 cycles per sec, what
is the percentage error in the record? Assume that the bridge and the oscillo
graph are properly calibrated by static tests.
1.23. Let the signal input to a galvanometer be
x s (t) =0 t <
x s (t) = Kt t >
in which A' is a constant.
(a) Find the expression for galvanometer response x r (t), assuming that the
galvanometer is calibrated for static readings.
(b) Show that the error x r — x s approaches a constant value.
(c) What would youestimate to be the optimumdampingratio in this galvanom
eter operation?
1.24. The indicial response function u(f) may be represented by
a n u(t) = 1  e(t)
where e(t) satisfies
1(e) =0 t >
ande(0) = 1, e'(0) = • • etc.
(a) Use this approach to derive (69b) by the use of (19) and (23).
(b) Use the same approach to find u(t) of a springmassdamper system for
cjc c > 1 and c/c c = 1.
(c) Obtain the answer for (b) by letting co c = iw for cjc c > 1 and by letting
co c approach zero for c/c c = 1 .
1.25. Draw the phase trajectory of the system in Fig. 27 with the initial
condition
x(0) = x(0) = x
1.26. A conservative system consisting of n mass particles m lt m 2 , . . . , m n is
constrained to move with a single degree of freedom so that the spatial coordi
nates of the particles are difTerentiable functions of a single configuration variable
6, that is, x t = x0) Vi = Vi (d) z t = 2,(0); i = 1, 2, 3, . . . , n.
Show that the kinetic energy of the system is given by
T = \m(d)() 2
in which m is a function of 6 only.
1.27. A suspension coil galvanometer, whose movement constitutes a spring
mass system, has an undamped natural period of 3.6 sec and a sensitivity of
1 in. deflection per microvolt. If this galvanometer is used in a circuit having a
resistance equivalent to 64 per cent of the critical damping, and an emf of 1
microvolt suddenly appears across the galvanometer terminal, find
(a) the time required for the galvanometer to deflect 1 in.,
(b) the maximum overshoot,
(c) the timedeflection curve for the first 10 sec.
96
THEORY OF MECHANICAL VIBRATION
1.28. The idealized vehicle in Fig. 39 hits a bump on a pavement which is
otherwise level and smooth. Let the profile of the bump be represented by one
period of a cosine wave:
/ 2ny\ L L
x s = a\ 1 + cos— I ~2 <2/< 2
x s = elsewhere.
Assume that a and R are small in comparison with L and that c/c c = 1.
(a) Find the vertical motion of the vehicle.
(b) What is the maximum horizontal velocity u the vehicle may have in passing
over the bump without the wheel leaving the pavement?
1.29. A platform of weight w is supported by a spring k and a shock absorber
c. At / = the system is at rest and a weight W is suddenly placed on the
platform.
(a) At what position will the platform finally come to rest again?
(b) Using this position as the new equilibrium position of the system and the
displacement from this position as the variable of configuration x, analyze
the motion of the system.
ra.
y 2 k
w
a
Exercise 1.29
1.30. If the weight W in Exercise 129 is dropped onto the platform from a
height h, instead of being placed on the platform, what changes will you have to
make in the analysis? (Assume that the weight does not bounce on impact.)
What is the maximum force transmitted to the foundation in terms of W and h ?
(Assume cjc c = 0.5.)
1.31. What is the damping ratio the system in Exericse 129 should have if
the requirement is that
\x\ <0.02(W + w)lk
for t > 3 T where T = 2irVkgl(W + w).
1.32. Show that
F{r)h{t r)dr = F{t  r)h
J'
Jo
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 97
1.33. Find the transfer functions between x r and x s for the systems shown.
M
\,h
m
1
t
EI
c
i
x s = force on m, F
x r = bending stress at A
(a)
Manometer:
Filled length = L
Fluid density = p
Fluid viscosity = m
x s = air pressure p absolute
x r = manometer height
(b)
Exercise 1.33
1.34. If the general form of a transfer function is
T(s) =
a s m 4 a^™' 1 +
b s n + V n_1 + ••■*«
where a's and 6's are real, 6 ^ 0> and n > m show that
Re [rtfo),)] = *((*>,)
is an even function of co, and
Im [TXfev)] = /(co,)
is an odd function of to,. Furthermore
lim ^ < oo and lim ^ =
V0 to/
co _> co w /
98
THEORY OF MECHANICAL VIBRATION
1.35. A heavy disk with a diametric groove rotates in a vertical plane. A
small mass particle m slides without friction in the groove under the influence of
gravity force, centrifugal force, and the force of a linear spring in the groove of
constant k. The disk is driven by external means to rotate at a constant speed
(a) Show that if we call k\m = w 2 , for co f 2 < w 2 /2, the path traced by the
particle in space is a circle lying in the lower half of the disk and for
CO* > COS > CO
72 the circle lies in the upper half of the disk.
(b) What happens when co f 2 = co 2 /2 and co f 2 > co 2 ?
Exercise 1.35
1.36. It is known that
f
2ttI J e .
ds = e ai
in which c is any real number greater than the real part of complex number a
c > Re (a)
(a) Use partial fractions to show that
e pt — e yt
and
J_
2m
fc + ioo
Jc — ico (■
1
rc+100
e st
M
Jc — ico \S
 P)(s  y)
)
s + a
. e st ds 
(a + 0)e/»  (a + y)e**
[s  Ms  y) PV
where c is any real number greater than the real parts of both $ and y.
SYSTEMS WITH A SINGLE DEGREE OF FREEDOM 99
(6) Use the results in (a) to show that
_i_ r +<o ° g ds = 1
2tj7 J c _i oo S 2 + CO 2 ^ CO
sin cof
2tt/ J ci co s' + CO^ CO
and
2wi J c
C + 100 Q *
se
ds = cos co/
c— too s \ oy
where a> is real and c > 0.
(c) Derive (74) and (72) by (94) and the foregoing results.
(d) Derive the impulse response for (78) by (94) and the results in (a) and
compare with results given in Art. 1.17.
CHAPTER 2
Systems with Two
Degrees of Freedom
SECTION A. THEORY AND PRINCIPLES
2.0 Introduction
All of the systems studied so far contain only a single variable of
configuration, say x(t), and are therefore called singledegreefreedom
systems. When a system requires two variables or coordinates, say x^t)
and x 2 (t), to specify its configuration at any instant t, it is said to have two
degrees of freedom. 1 Such a system serves as a simple model for the study
of the general oscillation characteristics of systems with several degrees of
freedom. The purpose of this chapter is to introduce the reader to the
more general analysis contained in Chapter 3.
2.1 Free Undamped Vibration — A Model and Its Equation of Motion
Consider the system shown in Fig. 44. It consists of two masses con
strained to move in a horizontal line. The masses are connected to a
stationary frame and to each other by three linear springs.
Using the displacements of the two masses from their respective
equilibrium positions, x 1 and x 2 , as the coordinates, we have the following
equations of motion for the two masses:
Wl 11 X 1 = ^n^l ^12V C 1 X 2'
7^22^2 == ^22^2 ^12v^2 X l)
1 A detailed discussion of degrees of freedom and its associated number of coordinates
is to be postponed till Chapter 3. For the present an intuitive understanding of the
terms is sufficient.
100
(2.2)
or
SYSTEMS WITH TWO DEGREES OF FREEDOM
^2^1 T ^22^2 T \^22 ' ^12/'*' 2 == ^
101
0)
This is a set of linear homogeneous differential equations with constant
coefficients. In the absence of first order (or velocity) terms, it is known
that the solution consists of linear combinations of sine and cosine
HyH
mm
W7a
x x
AAAA
Figure 44
x 2
wJdm
*22
functions with four constants of integration to be determined by the four
initial conditions of the system, which specify the displacements and the
velocities of the two masses at t — 0. Standard methods for obtaining the
solution are available. However, we shall take a physical approach that
will help us to understand the problem better.
2.2 Principal or Normal Modes
Before proceeding with our analysis, let us recall that of the three
quantities that characterize the free vibration of a singledegreefreedom
system, only the frequency is an inherent property of the system, whereas
the other two (amplitude and timephase angle) are determined by how and
when the system is set into motion by external agents. Consequently, it
interests us here to find out if there is also some behavior mode that is
inherent in a system with two degrees of freedom and is independent of the
manner in which the system is put into motion.
Since the system is conservative, a periodic motion may be possible.
This leads us to ask the question, "Can the system vibrate in such a way
that the motions of the two masses are simple harmonic motions of equal
frequencies?"
Assuming that this is possible, we set as usual
x 1 = ?. 1 e'
0)t
(2)
102 THEORY OF MECHANICAL VIBRATION (2.3)
in which ^ and X 2 are complex amplitudes. The substitution of (2) into
(1) results in a set of linear homogeneous equations in Xy and /. 2 .
(ra n co 2 + k u + k 12 )X x  k 12 ?. 2 =
(3)
k x2 l x + {m 22 o 2 + k 22 f ^12)^2 =
It is possible to satisfy (3), hence (1), other than by choosing the trivial
solution X x = X 2 = 0, if and only if the determinant of the coefficients
vanishes; that is,
■m u aj 2 f k u + k 12 —k
—k 12 —m 22 oj 2 + k 22 + k
12
12
= (4)
This is a quadratic equation in a> 2 and is called the frequency or character
istic equation of the system. Its roots are determined only by the constants
of the system. Simple algebraic operation shows further that the two roots
of co 2 are real, nonnegative, and distinct. Hence we conclude the following;
(i) The system is capable of motions describable by the equations in (2).
where co is real, hence the motions of the two masses are simple harmonic
in nature and have the same frequency.
(ii) There are two possible values of co that satisfy the conditions
prescribed. These become the natural frequencies 2 of the system. To each
natural frequency there corresponds an amplitude ratio /u = Ag/A^
m n a) 2 + k n + k 12 k
12
LJ ■ ll ■ L<i X£ fK\
/C 12 ^22 W ~r "^22 • ^12
These ratios are also real. In other words, although the /'s themselves
are not determined by the system, their relative magnitudes are ; further
more, since the ratio between the two is real, the displacements of the two
masses corresponding to a given natural frequency are in phase (or 180°
out of phase if the ratio is negative).
2.3 General Solution
Therefore, we have seen that there are two basic ways in which the
system can vibrate; they are called the principal modes of vibration.
Each is characterized by a nature frequency co and a corresponding
amplitude ratio ju. These two modes can operate alone; and. since the
2 As before, we call co frequency for short, instead of the more exact term circular
frequency, when there is no ambiguity involved.
(2.3) SYSTEMS WITH TWO DEGREES OF FREEDOM 103
system is a linear system in which superposition rule holds, the two modes
can operate simultaneously yet independently. Let the two natural
frequencies be co' and co" and the corresponding amplitude ratios, p and
/u". A possible solution of the equations in (1) is
Xl (t) = X^e iM ' 1 + X{e iuj " 1
(6)
x 2 (t) = iiix'e 1 ™ 1 + vyv""
in which A/ and X{ are two arbitrary complex numbers. Using standard
representation, we have
x x {t) = C ' cos (oj't  a') + C " cos (a/7  a")
(7)
x 2 (t) = fi 'C ' cos {co't — a') + ^"Cq cos {oft — a")
in which C ', C ", a', and a" are the absolute values and the arguments
of 2/ and A/', respectively. Altogether there are four arbitrary constants,
or constants of integration, which may be adjusted to fit any initial con
ditions. Thus (7) is the general solution of the differential equations of
motion, since it satisfies the equations and has the necessary number of
constants of integration. To give expressions of these constants, in terms
of the initial displacements and velocities of the two masses, involves only
routine algebraic operations, which can be dispensed with here.
The use of the displacements from the equilibrium positions of the
masses as the variables of configuration is arbitrary though convenient.
Any pair of independent geometrical quantities, which specifies the
location of the two masses, can be used as the variables of configuration.
We call a pair of such variables the generalized coordinates of the system.
In particular, any two independent linear combinations of x x (t) and x 2 (t)
can be chosen as the coordinates.
Suppose we choose a set of coordinates p', p", to satisfy
X l = P + P P = J7, 7T
or (8)
„ „ „ (px 1  x 2 )
x 2 = pp + pp p =— —
(/*  11 )
Then, from (7),
p'(t) — C ' cos (co't — oc')
p"(t) = C " cos (co't — oc")
(9)
104
THEORY OF MECHANICAL VIBRATION
(2.3)
Thus p and p" are two sinusoidal harmonic functions, each of which has
its own amplitude, 3 frequency, and timephase angle ; they are called the
principal coordinates and represent the principal modes of vibration of the
system.
If we had prior knowledge of // and p" and chose at the very beginning
p' and p" as the coordinates, we would have obtained two differential
equations of motion for the system, with each equation having only one
\
\
r
1
\
1
p"
r
J *i
,
P'
«»
p2
+rn'— i
< — ■ — m"
— >
"I
< — n' 
<
n"
^*
Figure 45
variable. Since their solution is known to be given by (9), they would be of
the form (see Exercise 24)
p' + (co') 2 p' =
(10)
p" + (co")Y =
In this way the two degrees of freedom are mathematically separated.
Because of the lack of clairvoyance in the values of ju' and p" we cannot
take advantage of this transformation of coordinates from k's to /?'s at the
very beginning. However, the existence of a transformation that reduces
(1) into (10) is of importance to our later discussions.
A mechanical model for the separation of the two modes can also be
conceived. In Fig. 45 we have two springmass systems of natural frequen
cies a/ and to". Two weightless levers are attached to the masses, as shown.
For small oscillations the displacements of the masses are given by (9).
3 It will become evident that the amplitude of p's has no intrinsic physical meaning
because in defining p's arbitrary multiplication factors may be introduced.
(2.4) SYSTEMS WITH TWO DEGREES OF FREEDOM 105
Let X x and X 2 be two points located somewhere on the levers, as shown.
The displacements of these points are then
YYl ryi'
x 1 (t) = T p'( t ) + T p"(t)
Xz(t) = n  P '(t) n p"(t)
Evidently, by making
ri
n
—
" = m
m
these equations are equivalent to (7), except for a multiplication factor.
In other words, with proper choices of a)', to", rijm', and n'jm", the
free vibrations of the system in Fig. 45 can be made to simulate those of
any undamped systems having two degrees of freedom.
2.4 Formulation by Energy Consideration — A General Analysis for the
Free Vibration of Systems with Two Degrees of Freedom
The two differential equations in (1) were set up by considering the
dynamic equilibrium of the masses. An alternate approach is by consider
ing the conservation of energy in the system. This approach is often more
convenient for systems with complicated constraints because of the
following reasons. The first is that energy expressions, being scalar and
nonnegative, can be written down with less reference to the geometry or
the kinematics of the system. The second is that constraint forces, which
do no work, will never appear in the resulting equations, whereas they
often appear in the equilibrium equations and have to be eliminated. 4
The kinetic energy and the potential energy expressions for the system in
Fig. 44 are
T = iO"irV + ™22 f 2 2 )
V = 2V^ll a i T ^^12'^1^2 ~T ^22^2 )
in which
C ll == ^11 "I" ^12 C 22 = ^22 "I" ^12 C 12 = — ^12
The absence of the term x^x^ in the first expression is due to the particular
choice of coordinates. In general, for a linear system whose configuration
is specified by a pair of generalized coordinates q x and q 2 will have the
4 See examples in Art. 2.11.
^^m
106 THEORY OF MECHANICAL VIBRATION (2.4)
following quadratic forms for the expressions of its kinetic and potential
energies : 5
T = i(«itfi 2 + 2a 12 q 1 q 2 + a 22 q 2 )
v = K<Mi 2 + 2c 12 q ± q 2 + c 22 q 2 )
This statement is amplified in the next chapter. In the meantime, we shall
consider (11) as the definition of a linear system. Since the system is in
free vibration, the principle of conservation of energy requires
^(T + V) =
at
Upon differentiating (11) and rearranging the terms, we obtain
(fln$i + c n?i + *12#2 + c i2 a 2)qi + Oi2?i + c 12 q x + a 22 q 2 + c 22 q 2 )q 2 =
(12)
This relation must hold for all possible vibrations of the system and at all
times. However, the values of q x and q 2 at a given instant are not related
by dynamics to the rest of the terms. 6 For instance, the system can be set
into motion with arbitrary combinations of q 10 , q 20 , q 1Q , and q 20 . Hence
the coefficients of q x and q 2 in (12) must vanish independently, or
<hfi\ + c n q x + a 12 q 2 + c 12 q 2 =
fl 12& + C 12^1 + fl 2 2^2 + C 22^2 = °
This is the set of differential equations that describes the motion of the
system. 7 These equations contain the same set of coefficients as the energy
expressions. The a 12 terms represent the inertia coupling between the
coordinates, and the c 12 terms represent the elastic coupling. 8
By either of the two approaches illustrated, we find that the frequency
equation of the system is
— a 12 co 2 + c 12
2 I n —n ,.£
= (14)
— a 12 (o* + c 12 —a^oj* + c 22
or
(a u a 22 — fl 12 2 )co 4 — (a n c 22 + c n a 22 — 2a 12 c 12 )w 2 + (c n c 22 — c 12 2 ) =
5 The change of the notation from the aw's to the as and from the x's to the </*s is in
keeping with the common usage. The a's and c's are called the inertia constants and the
elastic constants, respectively. Note also the appearance of the term a 12 in the general
system.
6 In contrast, the displacement </'s produce spring forces that in turn produce accelera
tion <7"'s and are thereby dynamically related to the accelerations.
7 Those who are familiar with Lagrange's equation of motion can easily verify that
(11) leads to (13).
8 Note that the presence or absence of coupling terms depends on the choice of
coordinates. When no coupling terms appear, the coordinates become the principal
coordinates.
(2.4)
SYSTEMS WITH TWO DEGREES OF FREEDOM
107
This quadratic equation of co 2 can be solved in the routine manner.
However, a particularly interesting representation of its solution is given
below.
Let us define
\ x u 12
a =
co„ 2 =
'12
12
a
22
0)
12
«11
*12
[ 12
L ll
C 12
f 22
a
'22
12
22
(15)
• 2 2
'12
22
12
It can then be verified that the two roots of co 2 in (14) are
'2
= 21!
CO
CO
iKl 2 + "22 2 ) ± \W
CO
2\2
!
) 2 + 4K 2 2 ) 2 ]
(16)
This is the familiar expression for the principal stresses in plane stress
problems. Hence Mohr's construction, shown in Fig. 46, can be used here
as well. The reason for bringing Mohr's circle into this discussion is to
point out that there is an intimate mathematical connection between the
analysis of the principal modes of vibration and of the principal stresses,
although the problems are physically unrelated. 9 Incidentally, the ex
pressions in (15) will not look so complicated if the inertia coupling is
absent, or a 12 = 0. In that case we may take the system in Fig. 44 as a
model and find
22
777,
777,
'12
2 _
V
Substituting (17) into (5) yields
OT n W 22
V
772 2 2^2
0)i
'12
(17)
(18)
Vm^ W12 2 W22 2  c ° 2
From Fig. 46 it can be seen that the amplitude ratios corresponding to the
two principal modes can be represented by the tangents of the angles d'
and 0" indicated
where 6' + d" = 90°.
V "jd^ = tan 0'
Vt7? 22 A 2 "
= —tan
(19)
10
9 Another problem having the same mathematical connection with these two is the
one dealing with the principal moments of inertia about centroidal axes.
10 The negative sign stems from the fact that co is less than either co n or co 2 2
108
THEORY OF MECHANICAL VIBRATION
(2,5)
Thus in the absence of inertia coupling the principal modes of vibration
can be represented by two perpendicular vectors in a proper coordinate
system, as shown in Fig. 47. A more generalized orthogonality relation
Figure 46
Figure 47
ship also exists in systems containing the "inertia coupling" terms, as
discussed in Chapter 3.
2.5 The Use of Influence Coefficients
The elastic property of a system is often more conveniently expressed
by a set of quantities called the influence coefficients. The influence coeffi
cient y 12 is defined as the displacement of the system at point 1 owing to a
(2.5)
SYSTEMS WITH TWO DEGREES OF FREEDOM
109
unit force applied at point 2. 11 The coefficients, y n , y 21 , y 22 , etc., are
similarly defined. It is stipulated, however, that at a given point all dis
placements have to be measured in the same direction as the unit forces
applied at this point, although they may not be in the same direction of the
displacements and the forces at the other point. The meanings of the
words "displacements" and "force" are generalized later to cover other
things. At present, it suffices to use them in their restricted sense.
Figure 48
To illustrate the use of influence coefficients in vibration analysis, let us
consider a system consisting of a weightless beam supported in some way
and carrying two concentrated masses at points 1 and 2, as shown in
Fig. 48. Let q 1 and q 2 be the transverse displacements of the beam at the
location of the masses at the instant t. Taking the beam as a free body, we
see that the two masses exert inertia forces of m n q 1 and m 22 q 2 to the
beam. 12 These forces with the necessary reactions at the supports produce
q x and q 2 .
Hence
q x = y^m sl q 1 + y^m 22 q 2
q 2 = y ZL m 11 q 1 + y 22 m 22 q 2
By expressing the q's in terms of the q's, we have
(20)
m V&\ = ~ (722^1 + 7l2? 2 )
y
(21)
m
22?2 =(yil#2 + 721?l)
11 It is implied here that the system is supported in such a way that forces can be
applied to it without producing accelerations; otherwise, the influence coefficients
cannot be immediately defined. (See Art. 2.9.)
12 With the beam as the free body under consideration, the forces exerted by the
masses are opposite to the accelerations of the masses. The forces shown in Fig. 48
are in their positive directions consistent with the positive directions of the displacements.
The actual directions of the forces are in the reverse, but this fact is to come from the
solution of the problem and need not be assumed beforehand. This is very elementary,
although at times confusing.
^™
110
where
THEORY OF MECHANICAL VIBRATION
(2.5)
y =
7n 7i2
721 722
By comparing (21) with (1) and recalling the definition of the spring
constants c's, we see that
Ci, =
Co =
7_2_2
7
7l_2
7
C 12 —
7n
7
721
7
(22)
Incidentally, we have also verified Maxwell's wellknown reciprocal
theorem which states
7l2 = 721
The relationship between the influence coefficients and the elastic
constants of a system, as represented by (22), has certain symmetry. We
can easily verify that
c 22
m = 
in which
722
c =
7i2 = 
12
11
12
12
22
(23)
Thus in the language of matrix algebra the influence coefficients and the
elastic constants of a system form matrices that are inverse to each other.
The transformation of (20) into (21) shows that it is not often easy to
say what kinds of coupling exist between the coordinates by looking at
the differential equation. In this problem there is no inertia coupling
between q 1 and q 2 , even though both accelerations appear in both equations
of (20). This is obvious if we write down the kinetic energy expression for
the system and observe the absence of a q x q z term.
Equation (20) can be directly solved without converting the influence
coefficients into the elastic constants. Setting, as usual,
in (20), we have, after multiplying the equations by — 1/co 2 ,
1
^ll7ll  — 2 J^l + ("^2712^2 =
(^Il7l2)^l + (^22722  ^P 2 =
(24)
(2.6) SYSTEMS WITH TWO DEGREES OF FREEDOM 111
The frequency equation is then obtained by the condition for the existence
of nontrivial solutions, namely, the vanishing of the determinant
1
m ii7n 2 m 22/l2
"Wl2 ^22722 2
CO 1
= (25)
2.6 Rayleigh's Quotient
In dealing with systems having a single degree of freedom, it was found
that the natural frequencies can be obtained directly by equating the
maximum values of the two energies.
T = V
We shall now determine where this procedure will lead us in dealing with
systems having two degrees of freedom. Let us assume that the system
vibrates in one of its principal modes with
q 1 = X x cos (cot — a) q 2 = X 2 cos (cot — a)
Hence 13
T m = i^ViA 2 + 2a 12 V 2 + a 2 2^2 2 ) =/^ 2
V m = ifoA 2 + 2c 12 A 1 A 2 + c 22 k 2 2 )
2 V m (X x , X 2 )
(26)
(27)
Since V m and / are homogeneous quadratic forms of X x and A 2 , co 2 is a
function of the amplitude ratio ju = X 2 \X V Without knowing the correct
value of /u, co cannot be found by Rayleigh's method, and if ju is to be
found by the method previously discussed we shall have found co already.
The matter, however, bears further discussion.
Let us consider the ratio of the quadratic forms as a function of ju and
call it Rayleigh's quotient.
n , v Cn + 2ci2^ + c 22/ w 2
60) = rz : 2 ( 28 )
Q(ju) becomes co 2 when /u is the amplitude ratio of a natural mode; other
wise it is always positive and finite, since the energy expressions themselves
are always positive and finite for any finite values of A x and X 2 . It is there
fore bounded both from above and from below. If these bounds can be
established, they will at least give us an estimate of the natural frequencies
of the system.
13 Aj and A 2 are real quantities in this discussion.
™
112 THEORY OF MECHANICAL VIBRATION (2.6)
It is not difficult to reason out that a plot of gversus// relationship
will result in a curve having the features shown in Fig. 49. Rayleigh's
quotient thus has two stationary values, one a maximum and the other a
minimum. These can be located by setting
dfji
which is equivalent to
dQ dO
9IT ar 2 = (29)
Substituting (27) for £) m (29), we have
bq u,dv m wr dp
£M = 1/£^_ F _2_ = o
3A, p\ J dl, m dXj
dQ \ PdV m rr df
Since Qf= T„
^ = _L ( f °J^v °L =0
Multiplication by Qf* yields
d
^r(v m TJ =
^(V m T m ) = (30)
Note that these are not trivial identities, even though T m — V m = 0.
In combining (30) with (26), we have
(flu 2 2 + c u )h + (~a 12 Q 2 + c 12 )A 2 =
(0120 2 + *iaWi + (^ 22 e 2 + c 22 )h =
The condition for this set of homogeneous equations to be consistent is
that the determinant formed by the coefficients must vanish. This con
dition is the frequency equation (14) exactly.
Thus we have shown that the two stationary values of Rayleigh's quotient
are the natural frequencies of the system.
(2.6)
SYSTEMS WITH TWO DEGREES OF FREEDOM
113
Perhaps it is worthwhile here to add a physical model to our analysis.
Take again the system consisting of two masses and three springs. Let us
incorporate into the system some device, such as a gear train, that con
strains the motions of the two masses to a given displacement ratio. In
so doing we reduce the system to one with a single degree of freedom.
Equations (26), (27), and (28), however, are still valid, except that fx =
IJX 1 is now a given ratio and the frequency of the constrained vibration
Q(n)
^ jU =
_ A2_
Figure 49
is determined by it. In general, if the ratio imposed by the constraining
device changes a little, the frequency of the constrained vibration changes
accordingly, But, when the ratio imposed is such that the frequency is a
maximum or a minimum, c/Q/d/Lt = 0, a small change in the ratio produces
a change in the frequency which is of a smaller order. The various elastic
and inertia forces in the existing vibration are therefore in an equilibrium
by themselves without the help of the forces exerted by the constraining
device. Hence, if the constraints are removed, the vibration remains
unchanged, and it is one of the normal modes of the system.
We have shown that the natural frequencies of vibration can be found
by maximizing or minimizing Rayleigh's quotient. However, to carry out
the process in its entirety requires almost the same algebraic steps as those
involved in the standard procedure. The practical value of the analysis
just presented is then lost. To apply Rayleigh's method in a practical way,
the amplitude ratio corresponding to a given mode instead of being
determined exactly is only estimated. The frequency is computed from
Rayleigh's quotient by using the estimated amplitude ratio. This procedure
gives surprisingly good results because, as we have seen, near a natural
frequency of the system the value of Rayleigh's quotient is insensitive to
small changes in the amplitude ratio. (See Art. 2.12.) In systems with
114
THEORY OF MECHANICAL VIBRATION
(2.7)
two degrees of freedom the exact value of the natural frequencies can be
obtained without much computation. The practical utility of Rayleigh's
method and its modification is in systems with many degrees of freedom,
but it is more clearly illustrated when applied to a simpler system.
2.7 Vibration of Damped Systems
If a viscous damper of damping constant c is added between the two
masses in Fig. 44, the term c{x 1 — x 2 ) will have to be added to the first
equation of (1) and subtracted from the second equation of the set.
In general, if there are viscous damping forces in the system, they will be
proportional to q's or linear combinations of q's in a manner analogous
to elastic forces that are proportional to the </'s or linear combinations of
the </'s. By adding damping terms proportional to the velocities of the
g's, the equation of motion (13) becomes
fl ll?l + b ll<il + C ll?l + «12^2 + ^12?2 + C 12 q 2 = °
*12?1 + *irfl + C 12?l + «22?2 + ^22^2 + C 22 q 2 =
The method of solution is analogous to that of solving (13); that is,
by letting
q 1 = X x e st or q 1 = l x e iat
and
q 2 = X 2 e si or q 2 = X 2 e iat
in (32). This leads to the frequency or characteristic equation
a n s 2 + b n s + c n a 12 s 2 + b 12 s + c 12
a 12 s 2 + b 12 s + c 12 a 22 s 2 + b 22 s + c 22
which is a biquadratic equation in s or ia, hence possesses four roots:
s l9 s 2 , s 3 , and s 4 ; the solution of (32) is then
= 0
(32) 14
(33)
(34)
q 2 = X 21 e s ^ + A 22 e**' + hz e *
+ h^
(35)
in which the ratios A u /A 21 , A 12 /A 22 , ^ 13 /A 23 , and A 14 /A 24 are determined by
s l9 s 2 , s 3 , and s 4 , respectively.
012*1 2 + *>12*1 + C 12
[ 22
Si + u 22 Si + c 22
^21
*ii*i 2 + Vi + c n
^12
^12^2 ~i~ ^12^2 "1" C 12
^22
a n s 2 2 + & u j 2 + c u
etc.
^12^1 2 + ^12*1 + C ia
^22^2 ~i~ #22^2 i ^22
a 12 s 2 2 + Z? 12 5 2 + c 12
(36)
14 We assume here that b 12 = b 21 , a fact which is discussed further in Art. 3. \
(2.7) SYSTEMS WITH TWO DEGREES OF FREEDOM 115
The initial displacements and velocities of the system and the relations
in (36) determine the eight constants of integration in (35). If all the roots
of (34) are real, the ratios in (36) are real; and with real initial values of
displacements and velocities, all the A's are real. The resulting motion is
nonoscillatory or aperiodic, and the system is overdamped. Furthermore,
since the damping forces dissipate energy, the motions must decay with
time and the real roots of (34) must be negative.
If (34) has complex roots, they must come in complex conjugate pairs,
since all the coefficients in this algebraic equation are real. Let
s ± = s and s 2 = s
be a pair of such roots. Then
e S x t _ e 8t e §4 _ gSt
will also be complex conjugate to each other, and, in order to yield real
values of q 1 and q 2 , 15 the first two coefficients in each of the equations in
(35) must also be complex conjugates.
Aq = A. A^ 2 ^ A
If in (36) we let
A 21 == '"^ll =: '" ^"22 == 21 == ^"
The part of the solution represented by the first two terms of (35) is then
qi = Xe st + le u
q 2 = rle Ht + ~rke gt
Let
s = ia = io) c — y\ X = \Ce~ x% and r — /ue~ id
Then
s = —ko c — 7] 1 = \Ce ly  r = jue l()
Substituting these into the expressions for q 1 and q 2 , we have
q x = Ce~ r,t cos (co c t — a)
q 2 = juCe~ 7lt cos (w c t — a — 6)
If (34) has two pairs of complex roots, the solution (35) may be expressed
by
q x = Ce' 1 ' 1 cos (co c 't  a') + Ce'^ 1 cos (co c "t  a")
q 2 = fi'Ce 71 ' 1 cos (co c 't  a'  6') (37)
+ ffC'ef' cos (co e "t  a"  6")
15 Note that (35) is a conventional representation for solutions of (32). Under any
real initial conditions it must yield real solutions.
116 THEORY OF MECHANICAL VIBRATION (2.8)
in which — r,' ± ico c ' and —rf ± ioof are the roots of (34); p\ fi", V,
and 0" are determined by (36); and C, C", a', and a" are determined by
the initial conditions.
It is evident from (37) that the oscillation of a slightly damped system
with two degrees of freedom may be decomposed into two characteristic
damped oscillations in a manner analogous to the decomposition of
undamped vibrations into principal modes. In other words, by choosing a
pair of coordinates p and p", which are certain linear combinations of
q± and q 2 , the set of equations (32) can be transformed into another set
in which the two variables are separated. The coefficients for this trans
formation are determined by (36) and will be complex numbers if the
roots of (34) are complex. The numerical computation involved in this
transformation is, however, very tedious.
To sum up, the nature of free vibration of a twodegreefreedom system
depends upon the roots of the characteristic equation (34). Since all the
coefficients of this equation are real, it may be factored into two quadratic
factors :
(s 2 + Irj's + to' 2 ){s 2 + 2rfs + o/ 2 ) =
The constant terms in the factors are the squares of the natural frequencies
of the undamped system because, if all the &'s in (34) are zero, r( = rf = 0.
and the equation is identical to (14). The two >/'s are the same as those in
(37). If there are damping forces, rf and rf in general will be positive,
although in special cases one of the two may be zero. To each of the
quadratic factors there corresponds a mode of vibration. In each factor
the condition ?/ < co indicates a damped oscillation with damped frequency
co 2 = (o 2 — ?y 2 , the condition >/ = indicates a sinusoidal oscillation,
and the condition r\ > to indicates an aperiodic motion. The condition
co = is discussed in Art. 2.9.
To factor the biquadratic equation, a numerical method must be used.
For such wellknown methods as Graejfe's rootsquaring method and LirCs
iteration method and its modifications the readers are referred to textbooks
on numerical analysis. It may be added that the general method of
solution for biquadratic equations is not convenient to use.
2.8 Forced Vibration
If a twodegreefreedom system is subjected to external excitation, the
differential equations of motion will no longer be homogeneous. The
external excitations may be forces applied to the mass particle in the
system or in motions imposed on certain parts of the system. In either
(2.8)
SYSTEMS WITH TWO DEGREES OF FREEDOM
117
case, the excitations are known functions of time, and the general form for
the differential equations of motion is
Ml + V/l + C ll<7l + fl 12?2 + ^12^2 + C 12 q 2 = QlO)
<*v&\ + &i2<7i + c i2<7i + a 22 q 2 + b 22 q 2 + c 22 q 2 = Q 2 (t)
(38)
This set of equations can be solved by a number of standard methods.
Here we will treat special cases of the most practical interest.
(A) STEADYSTATE VIBRATION UNDER PERIODIC FORCES
It is known that periodic functions can be expanded into a Fourier
series of sinusoidal functions. Since the system is linear and the principle
of superposition holds, the solution for periodic forces can be built up
from solutions for sinusoidal forces. We need therefore deal here only
with sinusoidal forces. Let
Qi(t) = V
and Q 2 {t) = F 2 e lu) f l
(39)
in which F 1 and F 2 are in general complex numbers. The steadystate
solution must be of the form
q 1 = ^e' 10 ' 1 and q 2 = l 2 e ni>lt
By substituting (39) and (40) in (38), we have
(— co f 2 a n + io) f b n + Cn)^ + (— co f 2 a 12 + uo f b 12 + c 12 )X 2 = F 1
( — co 2 a 12 + i(o f b 12 + c 12 ))^ + { — (» 2 a 22 + Uo f b 22 + c 22 )X 2 = F 2
(40)
(41)
This set of equations possesses a pair of complex solutions for ?. 1 and A 2 ,
which are to be interpreted in accordance with the discussion in Art. 1.7.
They are given by
F lt —oj f 2 a 12 + ia)fb 12 + ,c 12
F 2 , —co f 2 a 22 + ico f b 22 + c 22
(o 2 a u + ko f b n + c 11 ,F 1
—oj f 2 a 12 + icOfbji + c 12 , F 2
1
L
=2
D
1
h
=
~b
(42)
where
D =
co f 2 a n + ico fb n + c n , —co f 2 a 12 + ib 12 (o f + c 12
co f 2 a 12 + i(o f b 12 + c 12 , —«) f 2 a 22 + z^co/ + ^22
(43)
= (co/ — f^'coy — co' 2 )(o) 2 — irfcoj — co" 2 )
It is to be noted that if both rf and if are not equal to zero the determinant
D cannot vanish and X 1 and X 2 are always finite. On the other hand, if
118 THEORY OF MECHANICAL VIBRATION (2.8)
one or both of the rfs are zero and the determinant D vanishes for one or
two values of oo f , namely, to' and/or to", the correspondent / x and / 2 will
in general become infinitely large.
A frequency response curve for each coordinate in a twodegreefreedom
system can be obtained from (42) for any appropriate set of parameters.
It is not difficult to see that if the damping coefficients are not too large
all the curves will have two peaks near the natural frequencies of the
system. (See Fig. 58).
(b) analysis of an undamped system by principal coordinates
In Art. 2.3 (and Exercise 2.4) we have sketched a procedure whereby
the set of homogeneous differential equations (1) may be transformed,
through a linear transformation of coordinates, into another set of the
form (10). Such transformation is also possible for equations in the general
form (13) and for nonhomogeneous equations
%?i + Cntfi + «i2^2 + c 12 q 2 = Qi(0
(44)
<h!&\ + C 12<7l + «22^2 + f 22<72 = 2 (O
Without the use of matrix algebra, the steps involved become very tedious.
Since the subject is discussed more fully in Chapter 3, we give only the
final results here. If, as in (8), we let
ft = p' + p"
q 2 = p'p' + p" p "
the set of differential equations (44) can be transformed into
p> + osy = no
p" + yy = F\t)
where
(p!'a 22 + «i 2 )2i(0  (p"a 12 + a n )Q 2 (t)
HO =
HO =
(a n a 22  a 12 2 )(p"fx')
{pa 22 + fli 2 )8i(0 + (/^ /fl i2 + a u )Q 2 (t)
(a n a 22  a 12 2 ){fi"  p,')
(45)
(46)
(47)
Since the variables are separated in (46), the solutions for p and p" can be
obtained by methods discussed in Chapter 1. The initial conditions of/)'
and p" are determined by those for q 1 and q 2 through (45). Similarly, the
solutions for q x and q 2 are obtained by substituting the solutions for p
and/?" into (45).
(2.8) SYSTEMS WITH TWO DEGREES OF FREEDOM 119
Of particular interest are cases in which there is no inertia coupling
between the coordinates. For such systems a n = m n , a 12 = 0, a 22 = m 22 ,
and, according to (19),
' " . " hl n
ju /u H =
™22
The expressions of F' and F" can then be simplified to read
F'(t) = —JL Ql ( t ) + p'QJj)
m n(f*  P )
or
no = —t4 — k aw + /wo
m n {/4, — ix )
m n (/Li" — fif)(p' + co'Y) = fi'Q^t) + /x'ju' f Q 2 (t)
(48)
(49)
According to the principle of superposition discussed in Art. 1.11,
the solutions to (46) or (49) can be obtained by substituting the appro
priate parameters of these equations into (66), (74), and (76) of Chapter 1.
This method is particularly advantageous when transient solutions are
desired.
(C) INDICIAL RESPONSE FOR DAMPED SYSTEMS
When the system is damped, separation of variables by linear transform
is possible but not practical. Nonperiodic motion of a damped system
produced by external forces is of great interest in the study of control
systems. The problem is best treated by methods in operational calculus,
which is a subject by itself and cannot be included here. On the other
hand, the same problem is only of passing interest in theory of mechanical
vibration because complex systems with many damping forces are rare.
The simpler practical problems can usually be solved by more elementary
methods without too much extra labor.
If a transient problem is to be solved, the method discussed in Art.
1.11(d) may be used. First we obtain the transfer function in steadystate
analysis. For instance, according to (42), the transfer function from Q x
to q x or the frequency response of q 1 to Q x in (38) is
^ , x a ii s + b 22 s + c 22
Tjs)  — m — m
where D(s) may be factored into
D(s) = a(s  s x )(s  s 2 )(s  s 3 )(s  s 4 ) (51)
120 THEORY OF MECHANICAL VIBRATION (2.9)
in which a = a n a 22 — a 12 2 , and s t , s 2 , s z , and s 4 are the roots of (34).
If none of these roots is pure imaginary, the impulse response is then
^4f'S!^fH^ (52 )
ZTTlJicc D(S)
and the response <7 n (0, due to an arbitrary Q x (t) applied at t = when
the system is at rest and in equilibrium, becomes
fci(0 = 7~. " 2 " e^'^ ds) Q x {t) dr
LTTl JO \J»oo D{S) I
Should D(s) vanish for some pure imaginary values of 5. a real positive
constant c must be added to the limits of the integral with respect to s.
The impulse response function can be evaluated by the results in Exercise
1.35 after the numerator and denominator of the transfer function T(s)
are factored and the fraction is broken into partial fractions.
2.9 Degenerated Cases
(a) the case of zero natural frequency
semidefinite system
In most vibratory systems the vibrating masses are connected with a
stationary frame by elastic and constraining forces in such a way that the
equilibrium configuration of the system is uniquely determined and the
system can remain at this configuration only when there is no motion.
There is, however, a class of systems to which this does not apply. The
inertia elements in these systems may move as a rigidbody ensemble
without disturbing the equilibrium of the forces acting upon them. Such
systems are said to be semide finite. A simple example of a semidefinite
system is the one obtained by removing the two outer springs k n and k 22
in Fig. 44. This system may be considered as a model of such practical
things as two railroad cars elastically coupled or two pulleys mounted on a
rotating shaft.
To analyze the vibration of a semidefinite system, two approaches are
possible. We can treat it like any other vibratory system by writing down
the differential equations of motion and obtaining the frequency equation.
In the process of doing so we will find two peculiarities; the first is that the
influence coefficients of a semidefinite system are not all defined, and the
second is that the frequency equation will have a zero root, or roots, which
corresponds to the possible rigidbody motions. The difficulties introduced
by these peculiarities are discussed more fully in Chapter 3. The other way
of analyzing a semidefinite system is by eliminating the rigidbody motions
(2.9)
SYSTEMS WITH TWO DEGREES OF FREEDOM
121
with the assumption that the total momentum vectors (linear or angular)
associated with these rigidbody motions are identically zero. This is equi
valent to introducing constraints into the system and thereby reducing the
number of degrees of freedom. For instance, in Fig. 50 a rotating shaft
carrying three disks, representing a driving pulley, a driven pulley, and a
Figure 50
flywheel, has three degrees of freedom. One of the three is a rigidbody
rotation, which is of no interest to vibration study. The set of differential
equations is originally
/A + M&1  2 ) =
/A + k 12 (d 2  0j) + k 23 (d 2  6> 3 ) = o
I A + M^S  2 ) =
Adding the three equations together, we have
/A + hh + 1 A = o
(53)
or
/A + h®2, + ^A = constant
By assuming that there is no rigidbody motion, this constant becomes
zero, or
IA + I 2 2 + /
3^3
constant =
This last constant can also be set equal to zero as the 0's are taken to
be zero when the shaft is untwisted. By substituting
A#l + 7 3^3
®2= J
122 THEORY OF MECHANICAL VIBRATION (2.9)
into the first and third equations of (53), we have
i A + *„(«, + /A + /A ) = o
This set of equations can be reduced to the standard form of (13) by
multiplying them with the factors
h /^23 „ i h /^12
7 3 ^12 *1 ^23
respectively.
This procedure of eliminating a number of variables and equations
according to the number of possible rigidbody motions is, however, not
always convenient in a more complex system.
(b) the case of zero mass — "half" degree of freedom
We have seen that the motion of a singledegreefreedom system is
governed by a differential equation of the second order and that of a two
degreefreedom system is governed by two differential equations of the
second order, which are equivalent to one differential equation of the
fourth order. If, however, the inertia forces in one of the two degrees of
freedom are zero or negligible, the resulting differential equation of motion
becomes a thirdorder one. Consider the system shown in Fig. 51, which
has a single rigid body constrained to translate along a straight line. In a
sense this system has only one degree of freedom, since the displacement
function X(t) specifies the location of all mass particles in the system.
However, when the system is in motion, the value of X(i) does not specify
the location of the point P, whose motion must be considered as another
independent unknown. This system is in reality the degenerated case of a
twodegreefreedom system, which is obtained by first putting a mass m
at P and then letting this mass become zero. The differential equations of
motion degenerate into
MX + KX + k(X  x) =
ex + k(x  X) =
By differentiating the first equation to obtain the additional equation
MX + KX + kX  kx =
and by eliminating x and x from the three equations, we have
°^X + MX + (l + j)c* + KX=
(2.10) SYSTEMS WITH TWO DEGREES OF FREEDOM 123
This is a thirdorder differential equation whose characteristic equation is
— s* + Ms 2 +1 1 +t)« + a: =
This is a special case of the more general equation obtained by letting
a 12 = a 22 = in (34). Although the term makes very little physical
x(t)
k
X(t)
x/\/\/\/\/ N — ¥ — V\/\/\
K
Figure 51
sense, we may conveniently think of such a system as having one and one
half degrees of freedom. It is interesting to note that since a cubic equation
with real positive coefficients has at least one real negative root at least one
mode of the motion is an exponential decay.
X(t) = Ae~ st
2.10 Repeated Roots in Frequency Equations —
Transverse Vibration of Rotating Shafts
When the frequency equation (14) has two equal roots, the two modes of
vibration have the same frequency, and Mohr's circle, shown in Fig. 46,
degenerates into a point. The physical implication of this condition is
that any set of coordinates chosen to describe the two degrees of freedom
can be considered as principal coordinates in the sense that there are no
couplings between them. Consequently, such orthogonality relationship,
as described by (19), may or may not exist.
The analytical peculiarities introduced by the existence of repeated roots
in frequency equations are discussed in Chapter 3. At present, we shall
study a practical problem related to this subject.
Take the simple case of a circular shaft carrying a heavy disk in the
center and supported by bearings at the ends. Disregarding the rotational
freedom of the shaft for the moment, we note that the system has two
degrees of freedom which can be described by the displacement vector of
the disk center from its equilibrium position. Evidently the disk can be set
into a simple harmonic motion along any direction perpendicular to the
axis of the shaft. Hence any such vibration is a principal mode of vibration.
124 THEORY OF MECHANICAL VIBRATION (2.10)
For convenience, let us take the displacement components along two
mutually perpendicular directions as the coordinates of the system. The
differential equations for free vibrations are
mx + kx =
my + ky =
(54)
in which m is the mass of the disk and k is the restoring force per unit
deflection of the shaft at the location of the disk. The solution to this
set of equations is
x = x cos (cot — a)
y = y cos (ojt  p)
It is not difficult to show by analytical geometry that if ojt is eliminated
between the two equations the resulting relationship between x and y
represents an ellipse. In other words, the path traced by the disk center
is, in general, an ellipse. For the particular case in which x = y and
a — ft = ±7r/2, the path becomes a circle. The motion of the disk as a
whole is a translation. That is to say, the paths traced by all points of the
disk are congruent to one another, whether they are ellipses or circles.
For the sake of clarity in later discussions, let us coin the term whirling 16 to
denote the translation of the disk along a circular path. This whirling
motion may be excited by a force which is constant in magnitude but
which rotates with a uniform angular velocity co f . Let the force be in the
x direction at t = and have a constant magnitude F . Its x and y
components at any time t are then F cos co f t and F sin co f t, respectively.
When this force is applied at the center of the disk, the differential equations
of motion are
mx + kx = F n cos to J
(55)
my + ky = F sin co f t
According to Art. 1.6, for co f ^ co, the steadystate solution of (56) is
X = COS CO J
*[iK> 2 )]
p6)
y =
Fo 1
k [i  k> 2 )]
This solution represents a whirling motion. Let us observe that in this
motion the displacement of the disk center is always colinear with the
16 There is no standardized terminology for this motion. Some authors use the term
revolving or revolution for this purpose.
(2.10)
SYSTEMS WITH TWO DEGREES OF FREEDOM
125
force and that the two vectors are in the same or the opposite direction
according to whether co f is less or greater than co.
A rotating force such as the one described is realized physically when
the disk has an eccentricity and rotates with the shaft at the speed oj f .
Intuitively, it is reasonable to suggest that the rotation and the eccentricity
together produce a centrifugal force that is entirely equivalent to an
(a)
(b)
Figure 52
externally applied rotation force. To analyze the situation properly,
let us refer to Fig. 52. In this figure O is the position occupied by the
center of the shaft when the shaft is undeformed, C is the displaced
location of the shaft center, and M is the location of the center of mass of
the disk. The vector OC represents the translation of the disk and the
rotation of the vector CM, which has a fixed length equal to the eccentricity
e, represents the rotation of the disk. The acceleration of the mass center
M can be resolved into the relative acceleration of M with respect to C
and the acceleration of C with respect to O, which is fixed. The corre
sponding inertia forces are shown in Fig. 52a. In addition to these forces,
there is an elastic restoring force exerted by the shaft on the disk at C.
Assuming that at t — 0, CM is parallel to the xdirection, we have
mx + kx = mco f 2 e cos w f t
my + ky = mco f 2 e sin co f t
(57)
A comparison with (55) and (56) reveals that the displacement of the disk
center C is given by
x — r cos io f t
y = r sin ojA
126 THEORY OF MECHANICAL VIBRATION (2.10)
where
mco f 2 e oj f 2 e
k[\  k>2)]
(58)
and co f ^ co.
The disk therefore whirls and rotates at the same time. The shaft
center C whirls around O with the same angular velocity as the rotation
of the disk around C. The overall motion is a rotation of the disk about
the fixed point O. In other words the angle OCM is fixed. According to
our previous discussion, this angle is either or 180°, 17 depending upon
whether co f is greater or less than co.
From a practical point of view, for any co f that is different from a>, the
displacement vector OC can be made arbitrarily small by reducing the
eccentricity e. In other words, the whirling motion can be almost entirely
eliminated if the disk is very well balanced. However, as co f approaches to,
a very small eccentricity can still produce a large whirling motion if co f
is sufficiently close to co. This speed co is therefore called the critical
speed. The phenomenon is merely a special case of forced vibration at
resonance, and the critical speed corresponds to the natural frequency
of the system.
In engineering applications the rotating speed of a shaft is always kept
apart from its critical speed to avoid excessive vibration. In many high
speed devices the operating speed is above the critical speed. To reach
that operating speed, it is necessary during starting to pass through the
critical speed quickly. The behavior of the system near its resonance
condition is therefore also of practical interest. In Art. 1.6 it was pointed
out that if there is no damping steadystate oscillation is not possible at
o) f = co. Instead, the amplitude tends to build up as shown in Fig. 12.
In an actual case there is always some damping, and the amplitude
eventually becomes constant at some large but finite value. We also know
that regardless of whether or not a steadystate condition is reached the
excitation force leads the displacement by 90° at co f = co, so that the
relative location of O, C, and M is as shown in Fig. 52. Thus the centrif
ugal force of whirling motion, mco 2 r, and that of rotation, mco 2 e, are
perpendicular to each other. In addition, there is the elastic restoring
force kr along OC and another force F c along CM. This force F c may be
interpreted in two different ways.
First, let us assume that whirling has just started and that the damping
force is negligible, so that the amplitude, or r in this case, begins to grow.
17 If this fact is taken for granted beforehand, (58) can be derived by simple equi
librium consideration, as it is done in most textbooks.
(2.10) SYSTEMS WITH TWO DEGREES OF FREEDOM 127
This means that the angular momentum about increases; hence there is
an inertia force due to Coriolis acceleration.
F r r = — [mcoir 2 + e 2 )] = Imcor —
dt l dt
According to discussions in Arts. 1 .6 and 1 .9, drjdt = constant. Utilizing
(35) of Chapter 1, we have
dr d st mco 2 e coe
Jt = ~2 CO= ~2F° J= ^
Hence
F c = mco 2 e = ke
Or we can assume that the steadystate condition is reached by having an
amplitude large enough to produce a damping force
F c as ccor = mco 2 e = ke
Between these two extreme cases both types of forces exist, but they will
not be at their maximum values. Since the forces shown in Fig. 526 are
the only forces on the disk. Newton's second law demands that
F c = ke = ccor + Imcofr
These forces, however, do form a pure moment that must be balanced by
an external driving torque on the disk in order to maintain the condition
o) f = co. This torque is
M t = F c r = ker
Now if we want to speed up the shaft quickly, we must have a driving
torque that is substantially greater than M t . This is, of course, not a
quantitatively useful statement, since e is usually unknown and r varies
with / before the steadystate condition is achieved, But it does point out
the importance of having sufficient driving torque to speed up the rotation
quickly before large amplitudes can be built up to make the torque needed
for acceleration still larger.
In the foregoing analysis we assume that the only inertia force is that due
to a concentrated mass. This assumption is made merely to simplify the
illustration of the principle that the critical speeds of rotating shafts are
the same as their natural frequencies in transverse vibrations. If a shaft
carries several masses, it will have several critical speeds, which are to be
found by the method discussed in Chapter 3. If the mass of the shaft
128 THEORY OF MECHANICAL VIBRATION (2.11)
itself is not negligible, its critical speeds are to be determined by methods
discussed in Chapter 4. If the location of the disk on the shaft is such that
during vibration the disk also rotates about a diameter perpendicular to
the plane of the shaft deflection, then the rotatory inertia of the disk
about its diameter must also be taken into account. This effect is discussed
in Art. 2.14.
In short, if the entire system— the disks, the shaft, and the bearing
supports — has an axial symmetry, a rotating force can excite two trans
verse vibrations that are 90° out of phase and along two perpendicular
planes. The rotation of the shaft merely furnishes the excitation needed
without altering the vibrational characteristics of the system in other ways.
In other words, if the shaft is prevented from rotating while the disks are
allowed to rotate on the shaft, the result is assumed to be the same.
There are, however, circumstances in which the rotation produces excita
tion other than a centrifugal force that rotates with the speed of the shaft.
These are discussed in Art. 2.14.
SECTION B. METHODS AND APPLICATIONS
2.11 Illustrative Examples
In most cases the differential equations of motion for a vibratory
system can be written down without much preliminary analysis. But when
there are forces of constraint, which must be eliminated, the preliminary
analysis may become somewhat involved. In this article we illustrate the
procedure by analyzing two examples. The systems chosen have no
practical utility, but they serve our purpose well.
Consider first the system shown in Fig. 53. It consists of a wedge of
mass M sliding on a smooth table and a small block of mass m sliding
on the wedge. Two linear springs of constants k and K couple the two
masses together and to a stationary frame. Evidently this system has two
degrees of freedom, and a convenient choice of coordinates consists of the
horizontal displacement of M and the relative displacement of m on M.
Let us call these displacements x x and x 2 , respectively, and assign them
the value zero when the gravity forces are in equilibrium with spring
forces and forces between contacting surfaces. In this problem, the effect
of gravity does not change with x 1 and x 2 , and we need to consider only
the variations in these forces from their values at equilibrium configura
tion when we analyze the motion of the system. With this simplification,
the freebody diagrams of the two rigid bodies of the system are as shown.
The assumption is also made that these variations are small, so that the
(2.11) SYSTEMS WITH TWO DEGREES OF FREEDOM 129
string will remain taut and the block will remain on the incline. The
pertinent equilibrium equations are
TiF x = K{x x — x 2 ) — kx 2 = mx 2 — mx\ cos
SFtf = mx\ sin d = N
2F = K(x x  x 2 ) + Mx\  [K(x 1  x 2 )  kx 2 ] cos d + A^sin (9 =
(accelerations)
Figure 53
By substituting the first two equations into the third, we obtain
K(x ± — x 2 ) + Mx\ — mx 2 cos 6 + mx\ —
A rearrangement of the first and the last of the foregoing equations yields
(M + rri)x\ + Kx x — m cos Qx 2 — Kx 2 =
— m cos 6x\ — Kx 1 + mx 2 + (K + k)x 2 =
130 THEORY OF MECHANICAL VIBRATION (2.11)
This rearrangement reduces the equations to the standard form 18 of (13).
There are both inertia coupling and elastic coupling between the co
ordinates represented by the coefficients (— racosfl) and (— K), respec
tively.
Next we want to show how the same set of equations can be obtained
with much less labor and thought. It can easily be seen that the kinetic
energy and potential energy expressions of the system are
T = \Mx 2 + M(*i ~ * 2 cos df + x 2 2 sin 2 6]
V = \K{x Y  x 2 f + \kx*
By simplifying and rearranging the terms in these expressions, we have
T — \[(M + m)x^ — 2m cos 6x 1 x 1 + mx 2 2 ]
V = \[Kx 2  2Kx 1 x 2 + (K+ k)x 2 ]
A comparison with (11) reveals that
a n = M + m a 12 = — m cos 6 a 22 = m
c n — K c 12 = — K c 22 = K + k
By putting these constants into (13) and (14), we can write down the
equilibrium equation and the frequency equations immediately.
For our second example let us consider the system shown in Fig. 54.
A disk of mass M, moment of inertia /, and radius R rolls without slipping
inside a cylindrical surface of radius 3R. The disk is in vertical plane and
has a diametric slot in which a springmounted mass m can slide without
friction. This slot is at a horizontal position when the disk is at its lowest
point. The springs connected with m have an overall constant k, and the
neutral position of the mass is at the center.
This system contains many constraining forces in which we are not
primarily interested. Analysis by energy consideration will relieve us of
the necessity of dealing with these forces. Let us choose as the coordinates
for this system the angle 6 and the distance r, as indicated.
18 When the equations are obtained by equilibrium consideration, they are not always
in the standard form, but they can always be reduced to such form by simple algebraic
operations. It is most convenient to have the equations in the standard form because
their solutions can then be written down by direct substitution of constants into the
standard solution.
(2.1 1) SYSTEMS WITH TWO DEGREES OF FREEDOM
In terms of these coordinates, we have
v = velocity of the center of the disk = 2R0
Q. = angular velocity of the disk = 26
v m 2 = velocity of m squared
= (v cos 3d + ff + sin 3(9  rQ) 2
= v 2 + f 2 + r 2 Q? + 2i; r cos 3(9  2^ r Q sin 3(9
131
Figure 54
For small oscillations both r/R and 6 are small, so that the third and the
last terms can be neglected. After eliminating v , we have
v m 2 = 4R 2 6 2 + r 2 + 4Rf6
The kinetic energy of the system is then
T = iMv<? + ¥& + im^ 2
= i[4(M,R 2 + mR 2 + /)0 2 + 4m/?r0 + mf 2 }
Now let
?/ = the rise of the center of the disk = 2R{\ — cos 0)
= Rd 2
y m = the rise of m = y — r sin 2d
= R6 2  2rd
132 THEORY OF MECHANICAL VIBRATION (2.12)
The potential energy is then
V = Mgy + mgy m + \kr 2
= i[(M + m)Rgd 2  4mgrd + kr 2 ]
From these energy expressions it follows that the differential equations of
motion are
4(MR 2 + mr 2 + 1)6 + 2(Af + m)#0 + 2m£r  2m^r =
2mRd — 2mgd + mr + kr =
2.12 Application of Rayleigh's Method
(a) variation of rayleigh's quotient with amplitude ratio
Rayleigh's method is seldom used to analyze a twodegreefreedom
system, for which solution by the exact method is rather simple. To apply
this method to a simpler system does, however, give us the necessary
feeling of the method. First let us describe a method that is not really
Rayleigh's method. Take the system in Fig. 44 and assign the values
m n = 1 m 22 = 3
k u = 2 k 12 = 3 and k 22 = 1
all in their appropriate units. The expression for Rayleigh's quotient is
then
(k n + k 12 )X x 2  2k 12 X x X 2 + (k 22 + k 12 )X 2 2
Q(K A,) =
m^Xi + rn 22 X 2 2
5X 2  6X X X 2 + 4A 2 2
x x 2 + n 2
As a first trial, let X x = X 2 — 1.
56 + 4
0(1, 1) = — = 0.75
^ V ; 1 + 3
By substituting w 2 == 0.75 into either of the equations in (3) or (5),
we get some idea about how good our starting assumption is. In this case
we get
m n co 2 + k n + k 12 = 4.25 = ^ ^
k 12 3
The ratio X 2 /X 1 = 1.42 is most likely a better approximation. Rayleigh's
quotient for this ratio is
56 1.42 + 4 1.42 2 ^ ,^
W' 1A2) = . +31.42* =° 64 °
If we take this value to be a> 2 , we have co = 0.800.
(2.12) SYSTEMS WITH TWO DEGREES OF FREEDOM 133
The correct solution to three significant figures is co = 0.800 and
Ag/Aj = 1.45. To show how close Rayleigh's quotient gives the square of
the natural frequency when the assumed amplitude ratio is reasonably
correct, let us observe that
g(l, 1.3) = 0.652 co = 0.810
Q([, 1.6) = 0.651 co = 0.809
In other words, 10 per cent error in amplitude ratio produces about 1 per
cent error in frequency.
The other natural frequency can be obtained in a similar way. First we
know that co" 2 > co^ 2 = 5. If we assume that co 2 = 5, the second equation
of (3) gives Aj/Ag = — 1 1/3. (The first equation is too sensitive to changes
of co 2 when it is around 5.) Next we find Q(— 11, 3) = 5.67. Repeating
the process once, we find ^J^ 2 = —4.34 and co 2 = 5.69. These values
are nearly the correct answers.
In concluding this example, let us mention that the procedure demon
strated unfortunately cannot be adapted in a practical way to systems with
more degrees of freedom. The purpose of this example is to show that
when the amplitude ratio of a given mode of vibration is known approxi
mately Rayleigh's quotient provides the corresponding natural frequency
accurately.
(b) rayleigh's method for systems with known influence
coefficients
In contrast to the example just given, the following illustration has
practical utility because it can be immediately adapted to systems with
more degrees of freedom. The method to be discussed deals with systems
in which the influence coefficients instead of the elastic constants are
known. To apply the procedure previously shown, it would be necessary
to find the elastic constants according to relationships such as (22). This
is very inconvenient, especially for systems with many degrees of freedom.
A modification of the procedure is therefore needed.
Consider the transverse vibration of a beam supported at the ends and
carrying two concentrated masses weighing W n and W 22 , respectively, as
shown in Fig. 48. As usual, we start with the first trial.
K = h
In the previous example we determined or by Rayleigh's quotient
corresponding to the X l = A 2 and then obtained a better estimate of the
amplitude ratio by the equilibrium equations (3). From the revised
amplitude ratio the frequency is then obtained from Rayleigh's quotient
to a close degree of approximation. We now show that a revised amplitude
134 THEORY OF MECHANICAL VIBRATION (2.12
ratio can actually be obtained directly without computing Rayleigh's
quotient first. Let us write (24) as
h = Wl7l2^1 + W 22 y 2i ?.Jo) 2 lg
or
^1 = W ll7lA + F*Wl2*2 ...
/ 2 ^Il7l2^1 + ^22722 / 2
An iteration process can then be instituted by putting the estimated ratio
into the righthand side of the equation and obtaining a revised ratio from
the lefthand side. If our assumed firsttrial ratio is (/ 1: //. 2 )i = 1 tne
revised ratio is ,' , „ r , TTr 8
(h) = ^H7ll + ^22722 = £l (iij
U 2 / 2 ^ n y 12 + W 22 y 22 b 2
in which b ± and 6 2 are the static deflections of the shaft that would be
produced by the gravity forces on the weights if the beam were in a
horizontal position. Now we can compute Rayleigh's quotient corre
sponding to the revised ratio by letting k ± = b ± and / 2 = b 2 . The potential
energy of the elastic forces corresponding to these deflections will simply
be V m = «^iA + W 22 b 2 ) (iv)
and the value of V m is thus obtained without the need of converting the
influence coefficient into the elastic constants. Finally
n(h ,, Wi + ft, Ate M
If Aj = X 2 is anywhere near being a reasonable firsttrial ratio, (A 1 /A 2 ) 2 =
b 1 /b 2 will represent an improved estimate, and Rayleigh's quotient above
should yield the natural frequency very closely. It must be realized that
there is no inherent reason for AJAg = b 1 /b 2 always to be a good estimate.
For example, if the beam in Fig. 48 had a third pivot support between
the masses, the free vibration with the lower frequency would be such that
the masses would move in the opposite directions. If static deflections b ±
and b 2 are to be taken as the amplitudes, it is more reasonable to compute
them as if the gravity forces acting on the masses were in the opposite
directions. The results obtained when such artifice is needed are not
generally so reliable.
Generalization of the expression (v) to cover systems with more than
two masses is immediate.
WiA 2 + w^b* ■ ■ ■ w„„6
(vi)
(2.13) SYSTEMS WITH TWO DEGREES OF FREEDOM 135
Usually this approximation is good only for the vibration mode with
the lowest frequency, in which the static deflections more closely approxi
mate the amplitudes.
The appearance of the constant g in (ii) and (vi) does not mean that the
gravitational force has an effect on the natural frequency. If we were to
perform our measurements on the moon, where g is much smaller, the
(5's would also be proportionally smaller. The resulting value of co 2 would
still be the same.
2.13 Some Principles in Vibration Control
Methods and techniques in vibration control and isolation are matters
in art as much as in science. As pointed out by Macduff and Curreri, 19 the
problems involved are psychological, economical, and analytical, and
their degrees of difficulty are usually in that order. Within the scope of this
book only the basic principles concerning theory of vibration can be
discussed. From these principles successful solutions of practical problems
may be carried out if one also possesses experience, ingenuity, and persuas
iveness. The last quality is needed by the consulting engineer to convince
the client that the methods proposed are proper or the results achieved are
satisfactory.
(a) removal or minimization of excitation forces
Ideally, the way of removing vibrations in machines is to eliminate or
minimize the sources of excitation. In practice, this is not always feasible.
Much can be accomplished sometimes in the design stage of certain types
of equipment. Others, such as punch presses, highspeed card printers,
and riveting machines, have excitation sources that are inherent to their
functioning. In an existing machine, if avoidable and excessive sources of
excitation are suspected, measurements of the objectionable vibration
should be taken and analyzed. If the vibratory motion is found to be
clearly periodic, with a fundamental frequency forming a simple ratio 20
with the rotating speed or reciprocating frequency of certain parts of the
equipment, the source or sources of the excitation may be located and
sometimes removed by proper balancing and replacement of faulty parts.
When possible, efforts should also be made to vary the running speed and
to observe whether amplitude "peaks" at a certain particular speed.
19 See Macduff and Curreri, Vibration Control, McGrawHill, New York, 1958,
p. 117.
20 The resulting vibration can be the harmonics or the subharmonics of the excitation.
Subharmonics are produced by the nonlinearity in the system and have frequencies
lower than that of excitation.
136 THEORY OF MECHANICAL VIBRATION (2.13)
Should that be the case, the trouble lies with resonance, not with large
unbalanced disturbances. The corrective measure is then to separate the
resonant frequency from that of disturbing forces. The kinds of objection
able but avoidable vibrations described in this paragraph need not happen
if proper design and manufacturing procedures are employed ; but they
do happen just the same.
(b) the lowering of natural frequencies and the effect of
couplings
A cardinal principle of vibration control demands that the frequency of
excitation be kept away from any of the natural frequencies of the system
in which vibration is to be minimized. In most practical cases this principle
also demands that the natural frequencies be kept low. In springmounted
machinery springs with low stiffness will have to be used to keep the
natural frequencies low. Soft springs, however, pose practical problems
of their own. For instance, helical compression springs of low stiffness
may be too unstable without lateral guides. Also, the large static deflec
tions associated with soft springs may also become undesirable when the
total weight of the mounted system is a variable. For example, if the
suspension of an automobile is too soft, the road clearance may change
drastically as passengers enter. 21 Apart from the stiffness of the springs
used in mounting, their placement may also affect the natural frequencies.
Consider the system shown in Fig. 55, which may be taken as a simplified
representation of a long narrow machine mounted on springs. Let us
assume that the springs are guided so that the possible motion consists of a
vertical displacement of the center of gravity y and an angular displace
ment 6. For small oscillations the energy expressions for the'system are
T = \{Mf + Id 2 )
V = i[k(y  adf + k(y + bOf]
= \[2ky 2 + 2k(b  a)yd + k(a 2 + b 2 )6 2 ]
Using the notations in (17), we have
2k . k(a 2 + b 2 )
CO
11*  jjj 22 J
CO, J =
k(b  a)
12 " Vm
Evidently, if the springs are located so that a = b, there will be no coupling
21 A desirable feature of torsion bar springs and of air springs is that the road clearance
can be readjusted by an automatic device after the loading or unloading of the vehicle.
(2.13)
SYSTEMS WITH TWO DEGREES OF FREEDOM
137
between the coordinates, and the natural frequencies are co n and eo 22 .
The difference between a and b creates a coupling, and, according to (16),
this coupling has the effect of separating the two frequencies and making
the higher of the two still higher. Also, for a given distance between the
springs an increase in the difference a — b tends to make co 22 higher, since
2(a 2 + b 2 ) = (a + bf + (a bf
The existence of couplings between coordinates is therefore undesirable
in our effort to keep all of the natural frequencies low.
CG«
J
k §:
a —
h
W//////S
Figure 55
In discussing the effect of couplings on the natural frequencies, a
conceptual point must be cleared up. In Art. 2.4 it was pointed out that for
a given system the existence and the nature of coupling are matters
dependent upon the choice of coordinates, whereas the natural frequencies
are properties independent of the choice of coordinate system. For
instance, in the system just studied, \iy denotes the displacement of a point
midway between the springs instead of the center of mass, we will find
inertia coupling instead of elastic coupling. In the context of our present
discussion, when we speak of the effect of couplings upon the natural
frequencies we mean the effect of physical rearrangements that cause
changes in the coupling constants a 12 and c 12 between a given pair of
coordinates.
138
THEORY OF MECHANICAL VIBRATION
(2.13)
Generally speaking, the presence of symmetry favors the elimination
of coupling forces. When a piece of equipment is springmounted at the
base, it is usually possible to place the springs so that the vertical motion is
a principal mode. On the other hand, as long as the center of mass is
above the base support, any possible horizontal translation will be
coupled to rotary motions. (See Fig. 56.)
AAA/v
Figure 56
Aside from its effect on natural frequencies, a coupling between the
coordinates also couples the effect of external disturbances. In Fig. 55
an upanddown force acting on the center of mass will also produce a
rotary motion unless the two springs are properly located. If the natural
frequency of the rotary motion is near to that of the force, decoupling
them from each other will also produce a beneficial effect.
(C) DYNAMIC VIBRATION ABSORBERS
A dynamic vibration absorber is a small vibratory system coupled to
a machine or structure to control its vibrations. The absorber is designed
so that when the machine is subjected to a periodic excitation the resulting
vibration of the absorber produces a coupling force that tends to cancel out
the excitation force.
A simple form of this arrangement is shown in Fig. 57, where M is a
mass simulating a machine and ^is its mounting spring. The small mass m
and the coupling spring k constitute the absorber system. A damper c
may or may not be present. Assume that all motions are confined in the
vertical direction. Let Y be the displacement of M and y, the relative
displacement of m with respect to M. The energy expressions in terms of
(2.13) SYSTEMS WITH TWO DEGREES OF FREEDOM
these coordinates are
T=\MY*+\m(Y+yf
= i[(M + m) Y 2 + 2m Y y + mf]
V=iKY*+iky*
The power dissipation in the damper P is
P = cf
139
Let Q(t) be a force acting on M; the energy equation for the system is
evidently
d ,
~(T+V) + P=QY
From this equation and the energy expressions we obtain the differential
equations of motion:
(M + m)Y + KY+my= Q(t)
mf + my + cy + ky =
140 THEORY OF MECHANICAL VIBRATION (2.13)
This set of differential equations can also be obtained from equilibrium
considerations.
Suppose that Q is a sinusoidal force and that we are interested in the
steadystate response of Y. Let
Q(t) = Fe iu) f l and Y = leT*
According to (42) and (43),
F(—ma) f 2 + ic(D f + k)
1 =
D
D = [{M + m)aj f 2 + K](mco f 2 + k + icco f )  (nuo/f
= (K — Mco f 2 )(k — mco f 2 + icco f ) — ma) f 2 (k + icco f )
Evidently, if we design the absorber in such a way that kjm = a> f 2 and
c = 0, the amplitude of 7 will vanish. Hence, if the excitation is a
sinusoidal force with a fixed frequency, a simple and undamped dynamic
absorber "tuned" to that frequency will keep the vibration of the main
structure or machine to a minimum.
Although such simple absorbers are very effective when the excitation
force is at the designed frequency, they are of little value if oo f is subjected
to variations in a wide range. To prevent the amplitude of Y from
becoming excessive at all frequencies, it is necessary to introduce damping
in the absorber system. The following analysis for the optimum design
of a dynamic absorber is based upon the works of a number of investi
gators. 22 Returning to the general solution for A, we divide both the
numerator and the denominator by Kk and call
F A k 2 K H2 A CC °f
— 3= o e . — = co 2 — = £2 2 and — — = rj
K st m M k '
The nondimensional equation obtained is
X_
+ irj
m (co f \ 2 ,„
22 See Timoshenko, Vibration Problems in Engineering, Van Nostrand, Princeton,
N.J., 3rd edition, pp. 213220, and Den Hartog, Mechanical Vibration, McGrawHill,
New York, 1956, 4th edition pp. 93106.
(2.13) SYSTEMS WITH TWO DEGREES OF FREEDOM
141
To simplify the expression, set
(?)'•
(3)''
m A
 = y and
j = T(ico f )
(footnote 23)
a + /?y
d st (8(a + ty)  y(l  /SKI + A})
a + m?
~ (a0  y + y£) + i(P y + ypft
III" = ^±^ (i)
The object is now to design a system so that \T\ will be kept low for a
wide range of co f . Let us observe that, of all the parameters determining
j, tj is the only one containing the damping constant c, and that when
( g L py + yp \
a =±15 t (n)
\T\ is independent of r\. Physically, this means that for certain combina
tions of m, k, M, K, and co f the amplitude of vibration of M may be made
independent of the damping constant c. To determine such combinations
more specifically, we find that
= «g  7 + yP
Py + yp
is satisfied by a = 1 , p = 1 , or y = 0. These values imply the trivial
conditions m = or co f = 0. The other possibility
x= _ «ejy + yj}
P  y + yfi
leads to
(2 + y)aj8  (a  p + l)y =
which expands into
1 + W  (^ 2 + co 2 + yco 2 )u f 2 + w 2 Q 2 =
(iv)
This is a quadratic equation of co f 2 . It can be shown that the discriminant
of this equation is
(Q 2  (o 2 ) 2 + yo/(2 + y) >
23 There is no danger of confusing the symbol for transfer function with that for the
kinetic energy of the system. So we use T for both.
142
THEORY OF MECHANICAL VIBRATION
(2.13)
and the equation has two positive roots for oj f 2 . In other words, if a
family of frequency response curves (\T\ versus oj f ) is plotted with damping
constant c as a parameter, all the curves will have two points in common,
as shown in Fig. 58.
\T\
Conditions
7 = 0.05, g = 1
c x = 0.64 Wmk, c 2 = 0.20Vmfc
Figure 58
The abscissae of these two points, to x and co 2 , are determined by (iv),
and the corresponding ordinates 7\ and \T 2 \ are determined by (i) and
(iii). They are
1 7i = —T, — and rj = — ; — (v)
where
±(0i  y + rA)
*—(§)■
T ^ T(0 2  v + yh)
and
e) :
The signs to choose are the ones that make T's positive. It can be shown
that the correct signs are such that one is positive while the other is
negative.
To achieve an optimum design of the vibrationabsorber system our
approach is now as follows. We consider M and K as being fixed because
(2.13) SYSTEMS WITH TWO DEGREES OF FREEDOM 143
they belong to the system whose vibration is to be controlled. Tentatively,
we try to utilize the freedom we have in choosing three parameters, m,
k, and c, to achieve the following three things. One is to make IrJ = T 2 ,
and the other two are to make 7\ and \T 2 \ the maxima of the frequency
response curve. It is reasonable to expect that an optimum design would
be achieved when the ordinates of the two points, through which all
curves must pass, are made to be the two maxima having equal values.
Upon setting
\n = n
we have
(A  v + yA) = (&  y + yft)
Substitute in the expression for p x and /? 2 and simplify.
2H 2
V + co 2 2 = —  — (vi)
1 + y
Since co^ and co 2 2 are roots of (iv), their sum is
20 2 2(H 2 + co 2 + yw 2 )
This simplifies into
Or
1+y 2 + 7
n
(vii)
CO
1 + 7 ( vni )
m K / ra V
Affc \ + MI
This is one relationship that must be satisfied in an optimum design. The
corresponding value of \T\ is then
\t\ = rj = r, = Krj + r 2 )
Substituting (v) into the above, we obtain through simplification
ii ii 2coQ,
\ T i\ = Fa = 7—2 ^ lf w i > w 2
This expression evidently has to do with the difference between the
roots of the quadratic equation (iv), which may be written as
/coiy _ 2[Q 2 + co 2 (l + 7)] /oy>\ 2 _ Q
Uq; " (2 + 7)wft Uq/ 2 + 7
By utilizing (viii), we have
144 THEORY OF MECHANICAL VIBRATION (2.13)
Treated as an equation in co^/coH, the difference between its roots is
2 2  4 J^— =
2 + y N 2 + y
Hence
Z±l = J l+ ™ ( X )
y 'V m
and
con ± r
To make r a maximum at o, = co x and coy, = od 2 , we demand, if
possible, at oj f = a^ and co, = co 2
d(co f 2 )
For simplicity let us write (i) as
\T\ 2 (A 2 + #V) = a 2 + ?f
in which A = a/5 — y + y/3 and B = j3 — y + yfi, and use the symbol
prime (') to denote differentiation with respect to co f 2 . The condition (xi)
leads to
\T\ 2 (AA' + BB'rj 2 + £ 2 ??V) = oca' + ???/
According to (v) and (iii), at co^ and co 2
r 2 5 2 =l and ^ = a
Hence
#V = £aa' + v.A'
By utilizing the definitions of A and 5 and the relation
j8' co 2 V ^ 7;
we obtain through some simplification
(1 + y)rj 2 = a[(l  y)(B + p) + a + y] (xii)
Now the solution of (ix), together with (v), (viii), and (x), gives
% = dB) % = (i  m + r)
coll OT
< = (Ll9 and a = ±7^
Q 2 (1 + y) ^ r + 2
(xiii)
(2.13)
SYSTEMS WITH TWO DEGREES OF FREEDOM
145
Optimum conditions
#=1+7, 7 = 0.2
c 2 _ 3yk 2
optimum — 2 uQ
I
" 1
81
III
1°
in
1 °
~j
n
X.
2V
a
~1L
1
\\
i^
°1
Hi
/o
II
0.5
0.7
0.9
Figure 59
1.1
1.3
Elimination of a, ft, B, and r\ in (xii) by the use of (xiii) produces the final
result
(i)
2 W Q = 3y ± yVyl(2 + 7)
(xiv)
in which the "+" sign is for co f = c^ and the " — " sign for co f = co 2 with
a) x > co 2 . Since 7 ^ 0, it is not possible to make \T\ a maximum at both
* 46 THEORY OF MECHANICAL VIBRATION (2.14)
(0l and co 2 . Experience has shown, however, that if T versus the m t curve
reaches a maximum at one of the two co's, the other maximum has only a
slightly higher value. Hence a compromise can be made by taking the
average of the righthand side in (xiv). (See Fig. 59.)
©
2
. 3 mk 2 3m 2 koj 3 m 2 k
c i _
2 MojQ. ~ 2 MQ ~ 2 m + M
2.14 Effects of Rotation on Critical Speeds of Shafts
In Art. 2.10 the critical speed or speeds of a rotating shaft were con
sidered as nothing more than the resonant frequency or frequencies of the
system in transverse vibrations. It was maintained that the rotation
merely furnished the necessary excitation through the existence of
unavoidable eccentricity in the mass distribution of the system. In other
words, if we want to determine experimentally the critical speeds of a
certain shaft carrying one or more disks, we can, instead of rotating it,
excite it with sinusoidal transverse forces of different frequencies and
observe the speeds when resonant transverse vibrations take place.
The situation represented is a simplification that serves practical
purposes well in most cases. However, there are circumstances in which
the fact that the shaft is rotating has a bearing on the problem. In this
article we discuss briefly some of the effects produced by the rotation
itself.
(a) gyroscopic effect
In Art. 2.10 the inertia of the disk carried by the shaft is represented by
that of a single mass particle. This is permissible if the motion of the
disk during a transverse vibration is a pure translation. In general, the
motion of the disk also includes a rotation about an axis perpendicular to
the plane of bending, as shown in Fig. 60. The inertia of the disk thus
exerts a moment on the shaft when the system is in simple transverse
vibration. If the shaft is also rotating about its own axis, there is an
additional gyroscopic moment as the rotating disk changes its spatial
attitude.
Let us first investigate the case of a nonrotating shaft. Assume that the
system vibrates in the xyplane. Let y and d denote, respectively, the
deflection and the rotation of the cross section of the shaft where the
disk is located. During a transverse vibration in the xyplane the disk
exerts on the shaft a force my and a moment Ifi. When the system is in
(2.14) SYSTEMS WITH TWO DEGREES OF FREEDOM 147
simple harmonic vibration, y and 6 may be represented by
y = X cos cot 6 = j3 cos cot
At the extreme position of the disk, the inertia force and moment are as
shown in Fig. 60.
Figure 60
We now define the influence coefficients:
y f — deflection of a cross section per unit force applied at the cross
section
y m = deflection of a cross section per unit moment applied at the cross
section
(f) f = rotation of a cross section per unit force applied at the cross
section
(f) m = rotation of a cross section per unit moment applied at the cross
section
Maxwell's reciprocal relation is also valid here, so that y m = <j> f . With the
help of these coefficients, we can relate the deformation of the shaft with
the force and moment produced by the inertia of the disk:
X = moj 2 Xy f + I z co 2 Py v
p = mco 2 ?4 f + i>W,
(0
148 THEORY OF MECHANICAL VIBRATION (2.14)
This set of equations is evidently in the standard form of (24). The
natural frequency of the system is thus given by
m 7f i hYm
or
m 7m W* \
(ii)
It can be shown by simple algebra that this frequency equation has two
positive roots, oj x 2 and w 2 2 , and that co^ < co 2 < w 2 2 where co 2 =
\jmy t = k\m.
The transverse vibration of a nonrotating shaft with one disk thus has
two degrees of freedom in the x?/plane and another two degrees of freedom
in the xzplane, which are identical to those in the z2/plane.
Now, if the shaft is rotating, the system has five degrees of freedom.
The natural frequency in the fifth degree is zero, since it is a rigidbody
rotation. However, because of the gyroscopic effect a nonlinear coupling
exists between the rotation about the a>axis with that about the y and z
axes. A general analysis of the problem requires Euler's equation for the
rotation of rigid bodies. There is, however, a simple and practical situation
that we can analyze with elementary considerations.
Let us recall that the essence of the result obtained in Art. 2.10, as
expressed by (58), is that at co f 2 = kjm a shaft carrying a mass particle
with zero eccentricity is in a state of neutral dynamic equilibrium. It
can rotate in a bent shape as well as in a straight shape. For a real shaft
with a small but unavoidable eccentricity this speed marks a condition of
instability. The question is now whether the situation is still the same when
the mass carried by the shaft is in the form of a disk. Let us suppose
that the shaft and the disk rotate together around the xaxis in a bent
shape, as shown in Fig. 61. The centrifugal force acting on each mass
element of the disk is proportional to the displacement of that element from
the xaxis. This distributed centrifugal force produces a resultant force
perpendicular to the a>axis and a resultant moment which tends to restore
the disk to its neutral position. A comparison of Fig. 61 with Fig. 60
reveals that the moments of the inertia forces have opposite directions
when the system is in transverse vibration and when it is rotating around
the xaxis in a bent shape.
To determine the force Fand the moment M in Fig. 61, we can integrate
the contributions of each mass element of the disk with the help of an
appropriate coordinate system. The procedure is routine but a little
tedious; therefore we shall analyze the problem by a different approach.
The motion of the disk is assumed to be made up of a whirling motion,
(2.14)
SYSTEMS WITH TWO DEGREES OF FREEDOM
149
which is a translation, and a rotation about an axis parallel to the xaxis.
The translation produces a linear momentum whose time rate of change
results in a centrifugal force F given by
F = moo f 2 k
The rotation produces a moment of momentum L whose time rate of
change results in a moment M to be found as follows. Let us first observe
Figure 61
that the angular velocity vector co f , being along the a>axis, does not coincide
with any of the axes of principal moment of inertia of the disk. Therefore
the moment of momentum vector L is not in the direction of co f . (See
Fig. 62.) Let I x denote the principal moment of inertia of the disk about
its centroidal axis perpendicular to the plane of the disk. This axis is also
tangent to the deflection curve of the shaft. The component of L along this
axis, called L l5 is
L x = I^Wf cos ft)
Let I 2 denote the principal moment of inertia of the disk about the dia
metrical axis that lies in the plane of bending. The component of L along
this axis is
L 2 = I 2 (a> f sin fl)
150 THEORY OF MECHANICAL VIBRATION
The zcomponent of L is therefore
L x = L x cos p + L 2 sin ft
= (I t cos 2 + I 2 sin 2 ^cw,
(2.14)
= ^ + w, cos2 ^ W/ = 4w/
*l\^
cos
j^H
L ___ —
d
L 2 = I 2 oo f sin /3
Figure 62
and the ^/component of L is
L y = L x sin ($ — L 2 cos
= ( 1 2 2 sin 20 j w / = /„©,
in which 4 is the moment of inertia about the a>axis and I xy is the product
of inertia in the a^plane.
The socalled ^component, however, has no fixed direction, since it
always lies in the plane of bending and rotates with it. The time rate of
change of L is the same as that of L y and is represented by a vector in the
zdirection of a magnitude given by
dL v T T 2
— = oj f L y = I xy aj f 2
According to d'Alembert's principle, the moment produced by this change
is represented by a vector in the opposite direction, which gives the direc
tion shown in Fig. 61 through the righthand rule.
When f$ is small, this inertia moment becomes
M =
dt
 J> f = ft  hWP
(2.14)
SYSTEMS WITH TWO DEGREES OF FREEDOM
151
The minus sign indicates the fact that M is opposite /?. Using the influence
coefficient previously defined, we have
X = mo) f 2 ky f  (/i  I 2 )w f 2 py m
P = mco f ny m  (I ±  I 2 )a> f 2 p(f> m
(iii)
0)f COS
L\I x (co f cos /3  2w f )
L 2 I 2 u)f sin /3
Figure 63
Since this is a set of linear homogeneous equations in A and ft it admits
nonzero solutions only if
mv '~h
my,
~(h  h)Ym
co i
=
(iv)
For circular disks whose axial dimensions are small in comparison with
radial dimensions, we have
hh = h
and
h =
and
my f
(0,
i»y,
my,
W,
CO,
(v)
In comparing (v) with (ii), we see that the terms containing I z have different
signs. As a result, there are two positive roots for (ii) but only one such
root for (v). Hence for a shaft carrying a single disk there is only one
critical speed at which the system is in a state of neutral dynamic equili
brium. (See Exercise 2. 1 3.)
It has been observed in practice that sometimes the gyroscopic effect can
cause large vibrations in a rotating shaft at another speed, at which the
shaft whirls in one direction while the disk rotates in the opposite direction
at the same speed. A little reflection will reveal that the rotary motion of
the disk in this case can be represented by the sum of the two angular
velocity vectors, shown in Fig. 63. The ^/component of the moment of
152 THEORY OF MECHANICAL VIBRATION
momentum vector is then
L y = I^—lco f + (o f cos p) sin p — I 2 w f sin p cos ft
and for small p
(2.14)
Figure 64
The moment due to the inertia force is thus
M = (Ji + 7 2 )^ W/ 2
This time the moment is in the same direction as p. The rest of the pro
cedure is the same as before, and we obtain
™Vt 5 (A + / 2 )y,
CO
CO/

(vi)
This equation has two positive roots for co f 2 . The solutions to (iv) and (vi)
can be put into a common nondimensional form:
(S) 2 = \ [ ° + aIf ± V(l + a/) 2 + 4te/] (vii)
(2.14) SYSTEMS WITH TWO DEGREES OF FREEDOM 153
in which co 2 = \\mr f is the critical speed ignoring the gyroscopic effect
^™ JL 2
a = = 0m°>O
my,
and
v 2
6 = i __^_ (0<6< 1)
/ is the appropriate moment of inertia appearing in (iv) or (vi). The
hyperbola shown in Fig. 64 describes the relationship between c» f and /.
The relative magnitudes of the different roots of (iv) and (vi) are as
indicated.
Let us call the velocity condition described by Fig. 62 a forward whirling
and that by Fig. 63 a backward whirling. Figure 64 shows that forward
whirling can take place only at one speed, whereas the backward whirling
can take place at two speeds. It was assumed that in either forward
whirling or backward whirling the whirling speed is numerically equal to
the rotation speed. A question that may be asked is then, "Can the shaft
whirl and rotate at two different speeds?" Reviewing the method of
analysis employed, we see that given an arbitrary ratio of the whirling
speed to the rotation speed we can find, at least theoretically, the speed or
speeds of rotation at which the assumed ratio is realized. Since that is the
case, we must ask, "Why are the speeds given by (iv) and (vi) called
critical?" The answer to this question lies within the fact that all the
motions postulated are free motions in the sense that there is no energy
input. Because of the unavoidable dissipative forces in real systems, such
motions cannot be sustained without some kind of suitable excitation.
In the case of forward whirling at the speed of rotation the needed excita
tion is immediately available in the form of a centrifugal force produced by
the unavoidable eccentricity. The critical speed given by (iv) is therefore
always observable in practice, and it may be considered as the true critical
speed. The backward whirling cannot be excited as effectively by the
eccentricity, but a real shaft, which is a part of a piece of machinery, may
be subjected to other suitable excitation created by the rotation. Backward
whirling is therefore only occasionally observed. 24 Motions in which the
whirling speed and rotation speed are entirely different do not usually
have the necessary excitation to sustain them.
(b) effects of gravity force
If a shaft carrying a disk is in a horizontal position, the gravity force
will cause it to assume a bent shape even when it is at rest. But aside from
24 See Den Hartog, op cit., p. 265.
154
THEORY OF MECHANICAL VIBRATION
(2.14)
producing this change in the equilibrium configuration of the system,
the gravity force has no effect on the frequency of the free vibrations of a
nonrotating shaft. The situation, however, is different when the shaft
rotates in a horizontal position.
Consider the simple case in which the gyroscopic effect is not present.
During rotation there are three forces acting on the disk: a gravity force,
an inertia force, and an elastic restoring force, as shown in Fig. 65. In
Figure 65
this figure is the neutral position of the center of the shaft when it is
straight and 6 is the angle between the horizontal plane and the plane
of bending of the shaft. A normal way of operation is obviously that in
which the plane of bending remains vertical so that the gravity force
balances the elastic force and there is no inertia force as the disk rotates
about its lowered center M. The question to be asked now is, "Is it also
possible to achieve a dynamic equilibrium with the plane of bending
whirling around at a constant speed co f T Assume that this is possible.
Let us first resolve the inertia force into a radial component nuo f 2 r — mr
and a tangential component 2mco f f, which is the Coriolis force. The
whirling speed co f is assumed to be constant, and d = {co f t — a). Hence
mr + (k — mco f 2 )r = — mg sin (co f t — a)
(viii)
2mf(o f = — mg cos (co f t — a)
Obviously, these equations are satisfied by co f = 0, a = 77/2, and r =
constant = mgjk. Another way of satisfying (viii) is for co f to have the
or
(2.14) SYSTEMS WITH TWO DEGREES OF FREEDOM 155
appropriate value so that
r = r sin (co f t — a) (ix)
To see that this is true we substitute (ix) into (viii), and obtain
—mco f 2 r + (k — mw f 2 )r = —mg
2mco f 2 r = — mg
(k — 4mco f 2 )r =
oj f = \Vkjm = \coq
Since
where = co f t — a, the path traced by M as the plane of bending whirls is
a circle having radius d st and lying below but tangent to the horizontal
plane containing 0. 25
Theoretically, the motion described can take place regardless of the
speed of rotation of the disk. But again without suitable excitation this
motion cannot be sustained in real systems with the everpresent damping
forces. Tn real shafts the unavoidable eccentricity causes a whirling motion
at the speed of rotation. When this speed is cojl, the motion produced
by the gravity force reinforces that produced by the eccentricity, and the
resulting motion is represented by
• • x (0 o 2e
r = r sin (co f t  a) + —
>v
••6
— —
2S 8t
2mco f 2
k
X
=
r cos
= r sin
cos
= —
&8t Sil1
20
y
=
r sin
= r sin 2
d
=
<Ui
— COS
20)
. o)J \ 4e
— o af sin ,
1 2
With or without eccentricity, at the socalled critical speed of the second
order, co f = co /2, the vibrations produced have only finite amplitudes,
which are of the order of d st . Sometimes vibrations at this speed are
observed in practice with a substantially larger amplitude. Such vibrations
are usually due to other causes, such as a difference in the rigidity of the
shaft and its support in two different planes.
25 This motion is the same as one of those described in Exercise 1.35. It is, however,
instructive to compare the difference as well as the similarity between the system
described in Exercise 1.35 and the one being studied.
156
THEORY OF MECHANICAL VIBRATION
(2.14)
(C) HYSTERESIS WHIRLING
It has been observed in practice that a shaft rotating above its critical
speed may develop a whirling motion that is near its critical speed. This
phenomenon is commonly attributed to the existence of a hysteresis loop
in the shaft material. To explain such whirling motion on the basis of the
hysteresis characteristic of the material, we begin with a simpler pheno
menon with which the structural damping law of Kimball and Lovell, cited
in Art. 1.18, was first deduced. Consider a rotating shaft supported
between two bearings and subjected to an external lateral force P, which
causes the shaft to bend as shown in Fig. 66. If the shaft material is
perfectly elastic, the plane of bending will contain the force P, which
Figure 66
causes the bending. But if the shaft material has a hysteresis loop in its
stressstrain relation, as shown in Fig. 61a, the plane of bending will not
contain P. To arrive at this conclusion, let us assume that the plane of
bending is the f/zplane in Fig. 66. Consider a cross section of the shaft,
which is in the xyp\ane. The fibers above the xzplane, such as those
represented by points A and B in Fig. 61a, are under tensile strains,
whereas the fibers below (C and D) are under compressive strains. As
the shaft rotates in the direction indicated, it is not difficult to see that A
and C are being loaded, whereas B and D are being unloaded. Therefore,
if Fig. 61a represents the stressstrain history of all the outer fibers of the
rotating shaft, the stressstrain states of fibers A, B, C, and D are given by
those correspondingly labeled points in Fig. 67a. Hence the tensile stress
is higher at A than that at B, and the compressive stress is higher at C
than at D. If we consider all the fibers in the shaft, the net result is an
angle between the neutral axis of zero strain x — x and that of zero stress
n — n, as shown in Fig. 61b. Hence the external force P required to keep
the plane of bending in the ?/zplane must be making an angle a with the
2/axis, as shown. Let the deflection of the shaft where P is applied be
(2.14)
SYSTEMS WITH TWO DEGREES OF FREEDOM
157
A. The xcomponent of the force P and those of the reactions at the
bearing form a torque which must be balanced by an external driving
torque in order to maintain the speed oj f and the deflection A. The energy
input per revolution of the shaft is therefore
E* = 2ttPM\ = 2ttP sin a
(i)
This energy is dissipated by the hysteresis of the material.
As discussed in Art. 1.18, the experiments of Kimball and Lovell
established that
E t oc \Xf
Hence P oc A, and the angle a is a measure of the structural or hysteresis
(a)
Figure 67
damping force in the system. To summarize, if a rotating shaft is deflected
in a certain direction, it offers a resisting force which has two components.
One component is due to the elasticity of the shaft, and it is proportional
and opposite to the deflection. The other component is due to the
hysteresis of the stressstrain cycling suffered by the material in a rotating
shaft, and this component is proportional and perpendicular to the
deflection. It is 90° from the elastic component measured against the
direction of the shaft rotation.
As pointed out in Art. 1.18, strictly speaking, the description of the
hysteresis force given above holds true only if the deflection of a rotating
shaft is static. It is, however, reasonable to assume that this description
also holds when the deflection varies slowly with respect to time; that is,
the bending motion is slow in comparison with the rotation. Let us now
consider a shaft carrying an inertia mass and rotating at a speed above its
critical. In the absence of other forces the equilibrium configuration of the
158
THEORY OF MECHANICAL VIBRATION
(2.14)
system is such that the shaft is straight. Suppose that a transient distur
bance causes the mass to move away from the centerline of rotation.
Now we want to study the motion that ensues. The restoring force and the
inertia force on the mass are shown in Fig. 68. Let k and h be the constants
of proportionality of the elastic force and the hysteresis force, respectively.
The differential equations of motion are therefore
mx + kr cos 6 — hr sin 6 =
my + kr sin 6 + hr cos 6 =
(ii)
Centerline of
undeflected shaft
Figure 68
Since r cos 6 = x and r sin 6 = y,
mx + kx — hy =
my + ky + hx =
If we let £ = x + iy, the two equations can be combined to yield
ml + (k + /'/OS =
In the Argand diagram the complex variable £ graphically describes the
location of the mass m. Defining as usual h = i]k with i] = tan a > 0,
we have finally
ml + k{\ + irfrl =
or (hi)
ml + k£ =
where ac = &(1 + irj). The solution to (iii) is
I = 4^ + Be***
(iv)
(2.14) SYSTEMS WITH TWO DEGREES OF FREEDOM 159
where
1 = ±icoV\ + it} (v)
and (o = Vkjm = critical speed in the absence of hysteresis.
Through the usual complex number algorism, we obtain
S 2 \
For small ?] or ?/ <^ 1
and
i + Vi + v 2 ± / l + Vi +t i ) co
2 J 2
VI + yf = 1 + hf
s.
i
By substituting the expressions for s 1 and 5 2 into (iv) and remembering that
^ describes graphically in Argand's diagram the location of the mass in the
£?/plane, we conclude that the motion of m consists of a counterclockwise
whirling (+ico) with decaying deflection ( — fja>j2) and a clockwise whirling
( — ico) with increasing deflection (+rjcoj2). The latter motion represents
an instability of the system. Since the rotation of the shaft viewed from the
same plane is clockwise, the unstable whirling is with the rotation.
Returning to the physical picture, we observe that the foregoing con
clusions are contingent upon these assumptions:
(1) o) f > co; otherwise the experimental law (i) cannot be used.
(2) There are external driving torques sufficient to maintain the speed
of rotation m f .
(3) There are no other damping forces operating.
In an actual case a viscous force, such as air damping, will limit the
whirling motion to small amplitudes.
To close this discussion, let us mention one mathematical observation.
In Fig. 68, if we reverse the direction of rotation co f , the direction of the
force hr will also be reversed. Consequently (ii) and (iii) will be modified
into
mx + kx + /?«/ =
m V + ky — hx —
and
ml + Rt = (vi)
where R = k(l — irj) is the complex conjugate of k. Since (vi) can be
written as
ml + kZ =
160 THEORY OF MECHANICAL VIBRATION (2.14)
Its solution is complexconjugate to that of (iii). This time the unstable
whirling is in the counterclockwise direction, which is still the direction
of shaft rotation. Hence the physical picture remains the same. In other
words, whether we call k == k(\ + irj) the complex stiffness or R =
k{\ — irj), the result is the same.
(d) oil whip
When a shaft is supported by journal bearings with hydrodynamic
lubrications, the lift force produced by the oil film may pulsate and
interact with the elastic and inertia forces of the system to produce large
vibration. A detailed analysis of the problem is quite complicated. As a
first approximation, we may neglect the elastic effect and consider the shaft
to be perfectly rigid and straight. We know from hydrodynamic theory
of lubrication that in a journal bearing with a certain amount of clearance,
the journal is displaced from the concentric position, a lift force is pro
duced by the oil pressure distribution. This lift force is perpendicular in
direction and proportional in magnitude to the displacement of the
journal. 26 If a shaft is running with no transverse load and its journal is
displaced from the concentric position by a transient disturbance, this lift
force may cause the journal, hence the shaft, to whirl. The situation is
similar to the hysteresis effect already discussed. The equation of motion
may be written
mx f hy =
my — hx =
where h is found by Robertson 27 to be
h = H((o f  20)
with H a constant determined by the bearing dimensions and lubricant
property, co fi the rotation speed, and d, the angular velocity of whirling.
This set of differential equations can be satisfied by
6 = constant = co 1
X ) m t f C0S /
y) Urn
and
2ma) 1 2 — H(co f — 2co x ) =
In bearings of usual design H is often a large number in comparison with
26 If the displacement is small in comparison with the clearance. See Norton, Lubri
cation, McGrawHill, New York, 1942, pp. 106113.
27 "The Whirling of Shafts," by D. Robertson, The Engineer, Vol. 158, 1934.
(2.14)
SYSTEMS WITH TWO DEGREES OF FREEDOM
161
mco^ so that the whirling speed co x is very nearly equal to one half the
rotation speed co f :
Again we have a situation in which the displacement tends to infinity if not
checked by other forces unaccounted for here.
It may be noted that the two phenomena discussed in (c) and (d) do not
by themselves favor a particular speed or speeds to take place. Based upon
the much simplified analysis given, they will take place theoretically at all
speeds. In practice, other contributing factors determine whether or not
they will appear at all.
Exercises
2.1. Show that the solutions for co 2 in (4) are always positive.
2.2. In Fig. 44, if we define x x as positive when m lx moves to the right and x 2
as positive when m 22 moves to the left, what changes if any will we have to make
in (1), (3), (4), and (5) ? How will the values of co and /u be affected by this change
in sign convention ?
2.3. In general, are x ± (t) and x 2 (t), as described by (7), periodic functions?
Under what condition will they be periodic ?
2.4. Substitute (8) into (1), utilize (5), and compare the results with (10).
2.5. In Fig. 44 let m n and ra 22 weigh 2 lb each, k lt = 3 lb per in., k 12 = 2 lb
per in., and k 22 = 1 lb per in. Let the initial conditions of the masses be
2^(0) =2 in., x 2 (0) = 0, and ^(0) = x 2 (0) = 0. Find the motions of the two
masses.
2.6. A mass particle constrained to move in the x^plane is connected to a
stationary frame by three springs of equal natural lengths as shown.
3k
vwww
Exercise 2.6
162
THEORY OF MECHANICAL VIBRATION
►oo
zm
Kl
^•tto
AAAAA
WAAMA
SYSTEMS WITH TWO DEGREES OF FREEDOM 163
(a) With x and y indicated in the figure as the coordinates, write the differential
equations of motion for small oscillations of m and solve the frequency
equation.
(b) Without solving the problem, we know the two principal modes of vibration
are two mutually perpendicular rectilinear simple harmonic motions. How
do we know this is the case ?
(c) Find the directions of the two motions representing the principal modes.
(d) If we use Rayleigh's method to determine the frequencies, we shall need
a reasonable estimate of these two directions. The mode having the lower
frequency must favor the softer spring, and vice versa. So let us assume
that the lowfrequency mode is perpendicular to the spring 3k. Find the
frequency by Rayleigh's method and compare the result with those from
(a).
2.7. Find the frequency equations for small oscillations of the systems shown.
2.8. Find by Mohr circle construction the natural frequencies and the corre
sponding amplitude ratios of the two modes of free vibration of the system shown
in Fig. 53, if 6 = 30°, Mg = Img = 10 lb, K = 3k =15 lb/in.
2.9. Find and solve the frequency equation for the transverse vibration of a
cantilever beam carrying a mass m at its free end and another mass 2m at its
midspan. The length of the beam is L, and its section stiffness is EI.
<th
Exercise 2.9
2.10. If at the free end of the beam a spring of constant k = EI/L 3 is also
attached, find its frequency equation.
2.11. Prove (22) by the following considerations:
v = WA + /W
V = JfaA" + 2c 12 d^2 + c 22 S 2 2 )
in which d 1 and S 2 are the deflections at two points of an elastic system produced
by two loads P 1 and P 2 . Or
8 i = Pi/n + P2V12
d 2 = P\y\2 + JV22
2.12. Show that if the potential energy expression V is always positive for
arbitrary b x and 5 2 , which do not vanish simultaneously, it is necessary that
> and c xl c 22 > c 12
164
THEORY OF MECHANICAL VIBRATION
X^L
X
I
lL
yjmk ^2k
K a >+* a
(a)
(b)
m,I
(c)
Exercise 2.14
SYSTEMS WITH TWO DEGREES OF FREEDOM
165
Hint.
V = «( AA ± Vc 22 d 2 f T 2(V Cll c 22  c 12 )V 2 ]
V > for d 1 =
K > for d 2 =
K > for V^j ± V^<5 2 =
2.13. Show that there is only one positive solution to (iv) and (v) of Art.
2.14, since c n c 22 > c 12 2 implies y n y 2 2 > >"i2 2 
2.14. Set up the differential equations of motion in terms of the coordinates
indicated for the free vibrations of the damped systems shown and determine their
characteristic equations.
2.15. In the accompanying figure M represents an instrument which is packed
in a box B with springy padding having an equivalent spring constant K. Inside
VZZZZZZZZZZZZZZZZZZZZZZZZZZZ&
n\\\\s\\\\:
I
pagssgss ^
■M
^JEMJt
>zzzzzzzzzzzzzzzzzzzzzzzzzzzzl
Exercise 2.15
the instrument is a small part m shockmounted by spring k. The box with its
content is dropped from a height h. If
M weighs 25 lb \\K = 0.02 in/lb
m weighs 0.5 lb \jk = 0.5 in/lb
h = 5 ft
Find the maximum spring force in k.
2.16. An electric motor is mounted at the center of a horizontal beam that is
supported at its ends by springs and dampers and is presented from sideways
166
THEORY OF MECHANICAL VIBRATION
movement. An eccentric mass is attached to the rotor of the motor and is being
turned around in a vertical plane with a constant speed to f . Let
m = the combined mass of the assembly consisting of the rotor and the mass
attached
e = the eccentricity of the center of mass of the assembly
M = the combined mass of the assembly consisting of the beam and the stator
r = the radius of gyration of the assembly above
mmm
M2m I = Mr 2 r = Ae a = 2r
Exercise 2.16
Write the differential equations of motion for the upanddown and rotary
motions of the beam and find the steadystate solution.
Hint. Replace the rotating part of the system by a vertical inertia force, a
centrifugal force, a gravity force, and a torque.
2.17. Find an expression for the amplitude of the relative displacement x in
the steadystate operation of a dynamic absorber of optimum design in which
Q
= 1 +
m
M
and
r
3y
IcoQ.
2.18. Design a dynamic absorber under the following conditions:
M = 500 lb
K = 100 lb/in.
F = amplitude of disturbing force = 20 lb
m < 50 lb
D = coil diameter of the coil spring used for absorber spring
D < 4 in
S = shear stress in absorber spring S < 40,000 psi
2.19.
(a) Use the static deflection curve of the beam described in Exercise 2.9 to
find the lower natural frequency by Rayleigh's method.
(b) Do the same for the beam in Exercise 2.10.
2.20. A steel shaft 1 in. in diameter overhangs a long bearing by 4 in. It
carries at its free end a thin steel disk weighing 10 lb and having a radius of
gyration of 4 in. Find the critical speed of the shaft, taking into account the
gyroscopic effect due to forward whirling.
SYSTEMS WITH TWO DEGREES OF FREEDOM
167
2.21. In the system shown let the imput signal be the motion (displacement)
of the point P and the relative displacement between the two masses be con
sidered as the response :
(a) Find the transfer function.
(b) Find the impulse response.
wwwv
A/WVWV£
c = ^Jmk
Exercise 2.21
mmm
2.22.
(a) A simple pendulum consisting of a mass m and a light rigid rod of length L
is attached to the end of a vertical rod by a pin joint which allows rotation
in only one plane. If the vertical rod is spinning with a constant speed co f ,
describe the smalloscillation behavior of the pendulum.
(b) If the pin joint is replaced by a universal joint, analyze the same.
Universal
joint
Exercise 2.22
CHAPTER 3
Systems with Several
Degrees of Freedom
SECTION A. THEORY AND PRINCIPLES
3.0 Introduction
Having studied systems with two degrees of freedom, we can now
generalize our analysis to cover systems with any finite number of degrees
of freedom merely by incorporating additional coordinates into our
mathematical operation. With proper minor modifications, all the results
obtained in Chapter 2 can be applied to systems with several degrees of
freedom. There is, however, much to be gained by reexamining our
problem in its more general formulation and analyzing it with more
powerful mathematical tools.
In this chapter we shall write our equations in matrix form. Although
matrices by themselves can be considered mathematical entities, in
application to our problems we can consider them merely a kind of
shorthand. To help those readers to whom matrix notation is new, all
the essentials of matrix algebra needed for our purpose are outlined in the
Appendix. In the text we shall write our equations first in both the
"longhand" and "shorthand" notations and gradually turn to writing
in the shorthand notation only.
A few words are needed to defend the use of this mathematical tool,
which is not indispensable for solving practical problems. Three justifi
cations may be given. The first is that a simplified symbolism often provides
a clearer picture of the physical theory it represents ; a new formulation
may reveal new ways of solving old problems. For instance, numerical
168
(3.1) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 169
multiplication or division can be carried out with either Roman or Arabic
numerals, but the Arabic are not only more manageable, they also reveal
more about the nature of the arithmetic processes themselves. The second
justification is that matrix algebra is the unifying language of many
seemingly unrelated subjects. Its use enables us to borrow the experience
and feeling gained in analyzing one type of problem for the purpose of
analyzing another. And the third is that matrix notation has become
the accepted language for communication on the subject. Many technical
papers which deal with the topics discussed in this chapter are written in
this mathematical language. Furthermore, since the advent of electronic
computers, it has become increasingly necessary for engineers to communi
cate with the people who run the computers or to "communicate" directly
with the computers themselves. Both the operators and the computers
can understand and help to solve an engineering problem readily when
it is presented in matrix language.
3.1 Generalized Coordinates, Constraints,
and Degrees of Freedom
We begin our study with a more thorough discussion of the terms
mentioned in this heading. Any set of quantities, q v q 2 , . . . , q n , can be
considered as a set of generalized coordinates of a system if together with
a time t they can be made to describe the configuration of the system.
That is to say, when their values at any time t are known, the Cartesian
coordinates of every mass particle in the system are determined. These
generalized coordinates are said to be kinematically independent if the
geometry of the system does not impose any interdependence relationships
among them and their derivatives. In other words, it is possible for the
system to have any set of values of q's and q's at any time t. On the other
hand, if relationships such as
K(q v q 2 , • • • , q n ; q lf q 2 , • • • , q n ; t) =
exist, they are called the equations of constraint, or simple constraints of
the system. The number of degrees of freedom of a system is the number
of generalized coordinates, less the number of constraints among the
coordinates. There are two types of constraints. One type is called the
integrable constraint for which the function K contains no q's, or the q's
can be eliminated by integration with respect to /. For example,
K(q l9 q 2 ) = ft + 2? 2 =
or
K(q l9 q 2 , fa fa) = <7i?2 + q x q 2 =
170
THEORY OF MECHANICAL VIBRATION
(3.2)
With such constraints, some of the coordinates chosen beforehand can be
eliminated. If all the constraints of a system are integrable, it becomes
possible to have a set of generalized coordinates whose number is the
same as the degrees of freedom of the system. Such a system is called a
holonomic system and is the only type of system considered in this book.
As an example of a nonholonomic system, that is, a system with non
integrable constraints, consider the motion of a coin of radius r, which can
roll and spin without slipping on a rough horizontal table top. If it is
*~x
Figure 69
assumed that the coin plane remains perpendicular to the table top. the
configuration can be specified by the Cartesian coordinates of the coin
center, x and y, the bearing angle a of the coin plane, and the inclination
P of a diameter fixed in the coin, as shown in Fig. 69. The nonintegrable
constraint relationships among x, y, a, and ft are
rp cos a = x
rp sin a = y
This system therefore has only two degrees of freedom but needs at least
four generalized coordinates to specify its configuration.
3.2 Energy Expressions in Generalized Coordinates
for Linear Systems
We have seen in Chapter 2 that the vibratory motion of a system is
governed by a set of linear differential equations, if its potential energy is a
quadratic function of its coordinates and its kinetic energy is a quadratic
(3.2) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 171
function of the first derivatives of the coordinates. This is also true for
linear systems with more than two degrees of freedom.
T = K«ii?i 2 + a 22 q 2 2 + • • • + a nn q n 2 • • •
+ 2tf 12 ^ 2 + ^13^3 + 2tf 23 M3 + • • ') (1)
y = i( c n^i 2 + c 22^2 2 + * * * + c nn q n 2 • • •
+ 2c 12 ?i? 2 + 2c l3?l?3 + 2c 23^2^3 + " ') (2)
In fact, we shall define a linear system as one whose energy expressions
are as given above.
In addition to those systems that are genuinely linear many others can
be considered approximately linear if their motions involve only small
departures from their equilibrium configurations. To demonstrate this
fact let us assume in what follows that the configurations of systems are
determined by the generalized coordinates alone without having t entering
explicitly. The potential energy of a system, being an energy of con
figuration, is then a function of the generalized coordinates, the q's.
V = V(q v q 2 , ■ • • , q n )
Without losing generality, we may let q x = 0, q 2 = 0, . . . q n — 0, and
V(0, 0, ... 0) = 0, when the system is at its equilibrium configuration.
Thus the potential energy for any other set of values of the q's is, according
to the mean value theorem,
n [dV\ 1 n n I d 2 V \
Mi ( 3 )
in which the subscript of the first derivatives indicates that they are
taken at the equilibrium position (0, 0, 0, . . . , 0) and the subscript of
the second derivatives indicates that they are taken at some mean position
(Wu W2» • • • ' *7 A)> w ^ tn tne ^' s lying between zero and unity. 1
According to the principle of virtual work, when a system is in stable
equilibrium its potential energy is at a minimum. Hence the first derivatives
in (3) are zero. If we assume that the oscillation is small and the ^'s vary
1 We assume that V is twice differentiable with respect to all the </'s and the second
derivatives are continuous in the neighborhood of the equilibrium configuration. More
often one considers (3) as the Taylor series expansion, neglecting terms of third and
higher orders. In that case, the second partial derivatives will also be taken at </, =
and Fwill have to be continually differentiable.
172 THEORY OF MECHANICAL VIBRATION (3.2)
in small ranges, the second derivatives can be considered as constants. 2
Then V may be written
1 n n
v (<lv <l2>' ,q n )=22 CMi (4)
^ i j
where
Xdq.dq^
c H (i,j= 1,2, ••*,») (5)
This double summation in (4) when expanded is the same as (2).
For the kinetic energy expression let us consider a generic mass element
dm whose Cartesian coordinates x, y, and z are determined by the general
ized coordinates as follows :
x = x(q v q 2 , • • • , q n )
y = y(qi, q*, q n ) (6)
z = z (qv q* • > q n )
Its velocity in the ^direction, x, is then
n dx
* = Ip4i ( 7 )
i dq^
o ™n dxdx
x 2 = lZjr2qiqi (8)
Similar expressions for y 2 and z 2 yield the kinetic energy of the system.
T =  (x 2 + y 2 + z 2 ) dm
2 Jm
1 " * [ (dx dx dy dy dz dz \ i
2 » j hi\dq i dq j dq i dq j dq^q^
Again, for small oscillations, the partial derivatives can be considered
as constants with respect to / or the ^'s; and so is the last integral, being a
definite integral over a fixed amount of mass M.
Hence
T^lla.Mi (10)
2 Perhaps we are putting the cart before the horse, and should have defined the word
"small" to mean small enough so that the second partial derivatives may be considered
as constants.
(3.3) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 173
where
f (dx dx dy dy dz dz \
Again, this is a repetition of (1).
3.3 Summation Convention and Matrix Notation
Two conventions will be adopted for use in carrying out our analysis.
(a) the summation convention
We notice, in all the equations studied so far, that whenever a certain
subscript index is to be summed over all the available number of coordi
nates from 1 to n this index always appears twice in a given term. If this
is always the case, the summation sign conveys no additional information
and is therefore redundant. This is indeed generally the case for summation
processes arising out of the types of operations we deal with. Thus the
convention is hereby adopted that whenever a particular subscript index
repeats itself in a given term a summation process is understood, and no
summation sign will be used. Such an index is called a dummy index. In the
meantime, a nonrepeating index is called a free index. It can take on any
of the possible values 1, 2, . . . , n, but only one at a time. This convention
is intimately connected with tensor calculus. The resulting notation may be
called the tensor notation, although we may take it merely as a shorthand
notation. For example, in the equation
ft  c a a i
i is nonrepeating, hence, a free index, whereas j is a dummy index. This
equation is conventionally written
n
fi = 2 c «fc ( /= 1,2, •••,/!)
3=1
which represents a set of simultaneous equations written in "longhand 1 ' as
/i = c n<7i + c i2<7 2 + ' * ' + c ln q n
U = C 21?l + C 22?2 + * * ' + C 2n q n
fn = Cntfl + Cn2 a 2 + ' * ' + C nn q„
When an equation is written with the convention described, a free
index must appear precisely once in every nonzero term of the equation.
It is evident that the indexing denotes only a certain pairing or grouping
174 THEORY OF MECHANICAL VIBRATION (3.3)
relationship. Whether an index is named i or / or others is of no conse
quence as long as the proper pairing relationship is kept. For instance,
the following two equations are identical in every respect because they
are identical when written in longhand.
«< = c u<lj + bikPk
a k = c k m q m + KiPi
(b) matrix notation
Only three kinds of matrices are needed in our analysis. The notations
adopted are as follows :
(i) An n x n square matrix is denoted by a bold sans serif letter in the
upper case. The same letter in lowercase italic, with two subscript indices,
denotes an element in the matrix. For example, A represents a square
matrix, a 12 represents the element of A belonging to the first row and
second column, and a {j is a generic element of A.
(ii) An n x 1 column matrix is denoted by bold sans serif letter in
the lower case. The same letter in lowercase italic, "with a single subscript
index, represents an element of the column matrix. Thus a is a column
matrix, a 2 is the second element of the column matrix a. and a { is a generic
element of the column matrix a.
(iii) A bar above a bold sans serif letter denotes the transpose of the
matrix represented by this letter. Thus A is the transpose of A. and a
is a row matrix which is the transpose of the column matrix a. A row
matrix is always written as the transpose of a column matrix.
(iv) Greek letters with or without subscript and italic letters without
subscript denote scalar quantities.
(v) In this type of analysis, the number n, which specifies the order of a
matrix, is the same for all matrices within one problem. It is therefore not
necessary to be specific.
With the notation adopted, an equation written in matrix notation can
be readily translated into one in the tensor notation.
For example:
Matrix Notation Tensor Notation
a + b = ai + bi =
A B = C a u b jk = c ik
A i_ + g = aj] + gi =
P A q = A p i a ii q j = A
A B f + af = g "tfAi/ii + trfi = gi
Note that in writing the tensor equation corresponding to a given matrix
equation the last index of a matrix element must be the same as the first
(3.4) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 175
index of the following one. Similarly, to write the matrix equation from
its corresponding tensor equation, the dummy indices should be arranged
first to appear together. For example,
a thifi = d k
is first rearranged to give
hfiufi = d k
The corresponding matrix equation is
BAf =d
The same equation can also be written as
fiflifiik = d k
or
f AB = d
This ambiguity is not important because the two matrix equations represent
the same set of equations when written in longhand.
3.4 Free Vibrations of an Undamped System —
An Eigenvalue Problem
(A) DETERMINATION OF THE NATURAL MODES
In this article we shall carry out our analysis both in the tensor notation
(on the left) and in the matrix notation (on the right) and repeat some of
the more important results in longhand form.
The energy expressions in (4) and (10) can be written with the tensor
and matrix notations as
T = iq A q
~ (11)
K=iqCq
y = \c ij q i q j
The matrices A and C are called the inertia matrix and the elastic
matrix, respectively. They describe the inertial property and the elastic
property of the system.
In free vibrations the principle of energy conservation demands that
t + V=
t = KMA + <h&Ah I r = Kq A q + q A q)
Because both /and j are dummy Because t is scalar and A is
and a u = a H , symmetrical, the two terms in
the parentheses are equal.
t = a ij q i q i t = q A q (A = A)
176 THEORY OF MECHANICAL VIBRATION (3.4)
Similarly,
V=c ij q i q j  K=qCq (C = C)
The energy principle becomes
(arf, + CyqMt =  q(A q + Cq) = (12)
Since q t ; = or q = are trivial cases, we have
a itti + c {j qj =  A q + C q = (13)
In either form (13) represents the set of differential equations of motion
conventionally written:
Mi + c n?i + «i2& + Citfz + ' • ' + a ln q n + c ln q n =
a 2±qi + Cffltfl + «22?2 + C 22#2 + " ' * + tf 2n ?„ + C 2n ?„ =
a mqi + c nl ^ + a n2 q 2 + c n2 ^ 2 + • • • + a nn q n + c nB ^r fl =
The a's are called the inertia constants and the c's, the elastic or stiffness
constants of the system.
To solve this set of equations, assume as usual a typical solution:
q t = he i(ot  q = \e ia * (14)
and substitute (14) in (13)
(w 2 ^,. + cM =  (co 2 A + C) I = (15)
This is a set of linear homogeneous equations in /_,. The case that l t =
or I = represents the trivial case of no motion. However, these
equations can have nontrivial solutions; that is, if the determinant of the
coefficients vanish.
 c .. _ oj 2 ^ =0  C  co 2 A = (16)
Upon expanding the determinant, we obtain an algebraic equation in co 2
of the nth degree, which is called the frequency equation or the characteristic
equation of the system. It can be shown by theorems in algebra that since
T and V are nonnegative the frequency equation will have n nonnegative
roots. 3 Let these roots be
co 2 = co ± 2 co 2 2 , • ■ • , co n 2
They are the natural frequencies of the system. Let us assume for the
time being the commoner situation in which these roots are positive and
distinct. For each of the roots of co 2 it is possible to solve (15), in the
sense that it is possible to find the ratio among the /'s.
' 1 • '2 * ' 3 " * ' * " 'n = r l ' r 2 ' r 3 ' ' ' ' r n
3 See Whittaker, Analytical Dynamics, Cambridge University Press, 1937, p. 183.
(3.4) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 177
or
h = Wi I I = fir
where fx is an arbitrary complex number, and the r's form a column
matrix whose elements are real numbers determined to within an arbitrary
multiplication factor. The elements of r are real because the coefficients
of (15) are real. Therefore
q { = rtfie^  q = rfie™ 1
is a solution of (13), if (o satisfies (16) and r satisfies (15). The column
matrix r specifies the amplitude ratio of a natural mode of vibration.
It can be geometrically interpreted as a vector in flspace whose components
are the elements of the matrix r. Hence r is also called a modal vector.
Since fi is arbitrary, it is not necessary nor convenient to have r include
an arbitrary multiplication factor also. To remove this unnecessary
ambiguity in r let us adopt a socalled normalizing procedure, whereby
the first nonzero element in r is taken to be unity. In this way all the
other elements of r are uniquely 4 determined. From here on we shall
assume that the modal vectors are normalized.
The basic procedure for determining the natural modes of vibration
can thus be summarized as follows :
(i) Choose a set of appropriate generalized coordinates and obtain the
expressions for T and V, Lorn which the inertia matrix A and elastic
matrix C are determined.
(ii) Set up the frequency equation (16), using the inertia and elastic
constants of the system determined above, and find the roots of the
equation.
(iii) Put each root of the frequency equation into (15) and solve for
/ 2 , / 3 , . . . , /„ with I ± = 1. (If the solution is infinite with / x = 1, set / x =
and / 2 =1 ; then solve for the rest of the /'s. If this still does not give
finite answers, set the next element of / equal to 1 , and so forth.) A solution
obtained this way is the normalized modal vector r.
(B) GENERAL AND PARTICULAR SOLUTIONS
Let the modal vectors corresponding to the natural frequencies of a
system, oj v co 2 , . . . , co n , be represented by the symbols r (1) , r (2) , . . . , r (n) ,
respectively. The general solution of (13) is then
q = pCi)^"* 1 + r^fi 2 e iu> ^ + • • • r^fi n e im ^ (17)
4 We are assuming that the frequency equation has distinct roots. The case in which
it has repeated roots is discussed later. Also, the normalization procedure described is
one of the many possible. There is nothing physically unique about this procedure.
See also Art. 3.12(b).
178
THEORY OF MECHANICAL VIBRATION
(3.4)
where fA l9 ju 2 , . . . , p> n are arbitrary complex numbers which have to be
determined by the initial conditions of the system.
The expressions for this general solution can be condensed and brought
into more familiar matrix and tensor forms by introducing the following
two artifices. Let us first form a square matrix R, called the modal matrix,
by putting together all the column matrices r so that theyth column in
R has the same elements as r {j) . Or
' 10 ' I
U)
[ r <u
.(2)
,(ny
Then let us form a column matrix p whose elements are given by
P =
The solution can then be written simply as
Q i = r aPi I q = R p
?i(0 = >n/>i(0 + 'lW) + " " " + r ln p n (t)
ft(0 = *Wi(0 + r^pzit) + ■ • • + r 2n p n (t)
>r
>1^ V1
P2
=
ju 2 e iaj ^
.Pn.
jij^t_
(18)
(19)
(20)
<{JS) = r mPi(t) + r n2 p 2 (i) + ■ • • + r nn p n (i)
The set of functions p { (t) is defined either by (19) with complex number
representation or more conventionally by
P(')
pM
JPnV).
I//J cos (o^t — 04)"
\fi 2 \ cos (o) 2 t — a 2 )
\f*n\ cos (<o n t  a„)
(2D
The relationship between ju, \ju\, and a is
To obtain the particular solution satisfying a given set of initial condi
tions, these constants can be determined by the following procedure.
Let q(0) and q(0) be given. Put these into (20) to solve for p(0) and
and p(0).
qm = r ijP m
qM = r ti p0)
q(0) = R p(0)
q(0) = R p(0)
(22)
where p{t) is defined by differentiating (19) or (21). When p(0) and p(0)
are found, the constants ju, \/u\, and a can be determined from (19) or
(20).
(3.4) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 179
(C) EIGENVALUE PROBLEM
The method described for finding the frequencies and modal vectors
becomes impractical when the number of degrees of freedom n is large.
The number of arithmetic operations needed increases very rapidly with n.
To devise other methods, it is necessary to introduce a new concept,
the concept of an eigenvalue problem.
Stated in its most general form, an eigenvalue problem is a mathematical
problem containing an unspecified parameter in such a way that the
problem has only a trivial solution, unless this parameter takes on one
of a certain set of values called the eigenvalues or characteristic values of
the problem. The simplest and most important type of eigenvalue problem
is illustrated by the following example. Given a square matrix L, which
represents a linear transformation of vectors in flspace, we are to find a
vector r that is transformed by L into another vector in the same direction.
In other words, the transformation L changes only the length but not the
direction of r. Such a vector is called an eigenvector of L. The condition
to be satisfied is represented by the equation
Lr = Xr
The scalar quantity X represents the ratio between the lengths of the vectors
before and after transformation, and it has to take on one of the eigen
values for eigenvectors to be found. It is evident that eigenvectors are
determined only to within an arbitrary multiplication constant. However,
they can be normalized as desired.
The problem of determining the frequencies and amplitude ratios of the
natural modes of vibration can be stated as an eigenvalue problem.
Evidently the modal vectors satisfy (15)
(w 2 A + C) r =
This can be written
(A 1 C) r = co 2 r (23)
Hence the modal vectors and the natural frequencies are the eigenvectors
and eigenvalues of a linear transformation represented by the matrix
product (A 1 C). Or, L = A 1 C and X = co 2 .
The problem of finding the eigenvalues and their associated eigenvectors
is an integral problem. It is not necessary for one to be found before the
other. In the classical procedure the eigenvalues are found first. There are
other procedures in which the reverse is true or the eigenvalues and eigen
vectors are found simultaneously.
A matter of practical interest is the fact that finding the eigenvalues and
eigenvectors of a matrix has become a routine problem for electronic
180 THEORY OF MECHANICAL VIBRATION (3.4)
digital computers. In most computing laboratories the programming for
solving such a problem is readily available.
3.5 Principal Coordinates and Orthogonal
Property of Modal Vectors
(a) a transformation into principal coordinates
We have stated in Art. 3.1 that any set of quantities can be used as a
set of generalized coordinates of a system if by specifying their values the
configuration of the system is determined. It follows that ifq l9 q 2 , . . . , q n
are a set of generalized coordinates of a system any other set of quantities
that will determine the ^'s uniquely can also be used as a set of generalized
coordinates. According to (20) the set of quantities p ± , /? 2 , . . . , p n deter
mines q ± , q 2 , . . . , q n ; hence the /?'s can also be considered as generalized
coordinates of the system. Obviously, when the coordinates used to
describe a system are changed, the differential equations will be changed
also. Now we want to see how the equations change when the coordinates
are changed from q v q 2 , . . . , q n to p l9 p 2 , . . . ,p n . The matrix q satisfies
(13).
Aq + Cq = (13)
The matrix p then must satisfy
ARp+CRp=0
This equation may be premultiplied by (AR) _1 to yield
p + (A R)i C R p =
or
p + R i A 1 C R p =
Let us define a matrix W 2 as
W 2 = R^A^CR
Then p satisfies
p + W 2 p = (24)
In the meantime it is known that if q satisfies (13) and p is related to q
by (20) p l9 p 2 , . . . , p n are a set of sinusoidal functions satisfying
Pi + M i 2 Pi = °
p 2 + co 2 2 p 2 =
Pn + M nPn =
(3.5) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
This set of equations can be written in matrix form as
181
Pi
P2
+
Oh
.Pnl L ° ° '"CO,
Comparing this equation with (24) we have ;
[>1
"0"
P2
=
LPn_
_0_
R X A iC R = W 2
OJc
ft)„
(25)
In other words, W 2 is a diagonal matrix, and the transformation of the
coordinates from the q's into/?'s results in a separation of the variables in
the differential equations of motion. The /?'s are therefore called the
principal coordinates. 6
It must be borne in mind that the relationship (25) is one between a
matrix A 1 C, its eigenvectors which form the matrix R, and its eigenvalues
which form the matrix W 2 . It is a mathematical relationship not neces
sarily connected with the phenomenon of free vibration. For instance, in
a forced vibration problem we shall see that the matrix equation is the
nonhomogeneous equation
A q + C q = f (26)
The transformation procedure that leads from (13) to (24) transforms this
equation into
p + W 2 p=R 1 A 1 f (27)
in which W 2 is still a diagonal matrix and the unknowns p^p 2 , • • • ,p n
are separated in the set of differential equations represented by the fore
going matrix equation.
(b) orthogonality of modal vectors
A somewhat more convenient procedure of transformation into the
principal coordinates is based on the socalled orthogonality properties
5 We also have to take cognizance of the fact that them's are linearly independent in
order to arrive at (25).
6 If a certain normalizing procedure, such as the one described in the Art. 3.4, is used,
the principal coordinates become the normal coordinates.
182 THEORY OF MECHANICAL VIBRATION (3.5)
of modal vectors. Two vectors a and b are said to be orthogonal with
respect to a symmetrical matrix S if the product
a Sb = b Sa =
Now we want to show that any two modal vectors describing two natural
vibrations of different frequencies are orthogonal with respect to both
the inertia matrix A and the elastic matrix C. Let r U) and r (i) be two
modal vectors. They satisfy
cofA r«> = C r«
co/A r<>> = C r<«
co, =£ to, (28)
Premultiply the two equations by r 0) and r (0 , respectively.
wXr™ A r<*>) = r^ C r™
(29)
cofir™ A r (y >) = F (i) C r< J >
Because both A and C are symmetrical,
pW) C r (() = r (? '> C r (;)
pw) a r< ? > = F< ? > A r<«
Upon subtracting the two equations in (29) from each other, we have
(w, 2  cofiCf^A r<*>) =
Since oj t ^ co^ we have the proof that
fii) A r «) _
?wcpW0 "'*"' (31)
In tensor notation these two equations are written
r k a km r m — u ,
f\ t\ c °i ^ c °i
r k C km r m — U
In either way of writing the equations contain only scalar products
involving modal vectors. The same orthogonality property can also be
described in terms of the modal matrix R given by (18). Let us form the
matrix products
r k PkrJmi = ™,i R A R = M
r kfkm r mi — kji R C R = K
We can see from (18) and (31) that if the natural frequencies for different
modes are. distinct the matrices M and K will have only diagonal terms.
That is,
m {j = k tj = when / =£y*
(3.6) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 183
The transformation of coordinates from q's to /?'s can now be performed
on (13). Substitute (20) into (13) and premultiply the equation by R.
The result is
ARp+CRp=0
RARp+RCRp=0
or
M p + K p = (32)
Since M and K have only diagonal terms, the longhand expression for
(32) is
™iipi + k nPi =
m 22p2 + k 22 P2 =
m nn p n + k„ n p„ =
Evidently, since p is given by (19),
^n 2 22 2 7
m u «7 2 2 ;77 7
This can also be arrived at by the following operations:
M 1 K = (R A R)i(R C R) = R 1 A 1 R  1 R C R
= Ri A^C R = W 2 (33)
The transformation from (13) to (32) is called orthogonalization.
The mathematical problem of determining the natural frequencies and
modes of vibration is also one of finding a matrix R which will diagonalize
two given symmetrical matrices A and C through
RAR=M and RCR=K (34)
and will give the natural frequencies through
M 1 K = W 2 (35)
This approach is used in some numerical methods of solving the problem
by successive approximation.
3.6 Rayleigh's Quotient
In Art. 3.5 the two equations (34) and (35) imply that if r is a modal
vector the quotient
184 THEORY OF MECHANICAL VIBRATION (3.6)
is the natural frequency of vibration associated with that modal vector.
Let us now investigate the nature of such a quotient formed with an
arbitrary vector u. This quotient is called Rayleigh's quotient of u.
u C u
fi(u) = =— (36)
u A u
Let us retain the assumption that the frequency equation has distinct
roots. With this assumption, it is not difficult to show through the
orthogonality property of the modal vectors that these vectors are linearly
independent. (See Exercise 3.10.) An arbitrary vector u in «space can
thus be expressed in terms of a linear combination of the modal vectors
u = pjpff) + v 2 r {2) + * * ' + v n r {n)
in which v l9 v 2 . . . , and v n are a set of coefficients. By utilizing definition
(18), we can rewrite this equation in a form similar to (20):
Ui = r ijVj  u = R v (37)
The meaning of (37) is, however, slightly different from (20), since (37)
does not contain the time variable /.
Rayleigh's quotient formed with the vector u can thus be expressed
vRC Rv VKv
eoo =  BAP = =r— (38)
vRARv vMv
Since K and M are diagonal matrices and K = M W 2 , we may write (38)
in longhand form :
n  m uV ( °i 2 + ^22^2 2 ^2 2 + • • • + m nn v n 2 co n 2
t^ 2 i 2 i i 2 W"/
m^vf + m 22 v 2 2 H + m nn v n z
Consider Q as a function of n variables, i\, v 2 , . . . , v n , defined by
(39). Since the denominator in (39) does not vanish except when i\ =
v 2 = . . . = v n = 0, the function is continuously differentiable except at
the origin. Hence the function possesses a Taylor series expansion, and
for small changes in the y's the change in Q is given by
AS = IP &v t + I J!f A,, A., + • • • (40)
in which the summation convention is employed.
Let us assume now that u coincides in direction with one of the modal
vectors, say r (1) ; then u = v x r a) and v 2 = 0, i? 3 = 0, . . . , v n = 0. Through
routine differentiation of (39), it can be shown easily that under these
conditions
££ = e = 0,..,§e = (41)
ov 1 dv 2 dv n
(3.7) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 185
Similarly, these partial derivatives vanish when u coincides with any
modal vector r (i) . Hence Rayleigh's quotient achieves a stationary value 1
at any of the modal vectors. Furthermore, if 6o x < co 2 < ... < co ni w^
is the absolute minimum of Q. In other words, Rayleigh's quotient is
always larger than or at least equal to the lowest natural frequency of the
system. (See Exercise 3.11.)
By substituting (41) into (40), we obtain the relation
2 ov t ovj
which holds in the neighborhood of a modal vector. This relation indicates
that if the difference between a vector and a modal vector is small, small
changes in this vector will produce only secondorder changes in the value
of the corresponding Rayleigh's quotient. Since the value of Rayleigh's
quotient for a modal vector is the square of the natural frequency of the
corresponding mode, a vector that is a first approximation of a modal
vector will yield, through its Rayleigh's quotient, a better approximation
of the corresponding natural frequency. This approximate method of
finding the natural frequency is called Rayleigh's method.
3.7 Forced Vibration of an Undamped System
(A) GENERALIZED FORCES
Since the coordinate system used in this analysis is in a generalized sense,
it becomes necessary to have the corresponding concept of generalized
forces. Let us first be reminded that our analysis is based on the study
of energy forms, and our equations are derived from energy expressions.
It is therefore obvious that those external forces which act on the system
but do no work do not enter into the picture. Forces of this type are called
forces of (nondissipative) constraint. Forces whose work upon the system
is included in the potential energy expression are called potential forces and
need not be considered again. 8 Therefore, only the forces that do work
but are not accounted for in the potential energy of the system need to be
added to the equations of motion (13).
Since the consideration here is energy, it is natural to define generalized
forces by the work they perform on a system. The power, or the time rate
7 A stationary value can be a maximum, a minimum, a minimax, an hyperbolic point,
or others.
8 For example, the gravity force does work on a simple pendulum; but this work is
the potential energy of the system, and the force is taken into account in the change
in potential energy.
186 THEORY OF MECHANICAL VIBRATION (3.7)
of doing work, by external forces is a linear function of the velocity
components of the mass particles on which the forces are acting. According
to (7) the velocity components themselves are linear functions of the qs.
Hence the power of external forces may be written as linear functions of
the q's:
Mi = Mi + Mi + • • • + Mn (42)
The coefficients /'s are called the generalized forces with respect to the
coordinate system q t . For linear systems and for systems in small oscil
lations these coefficients can be considered as independent of the q's and
are functions of time t only.
With generalized forces acting on the system, the energy equation
becomes
t+V=f i q i (43)
By following the same steps in the derivation of (13), we arrive at
<*<A* + c iAi =fi I A q + C q = f (44)
(b) steadystate response to harmonic and periodic
generalized forces
The special case in which every f t is a harmonic function of time having
the same frequency co f is now considered.
Let the generalized forces be represented by
ft = gie""''  t = pf*
In this case the solution consists of a steadystate vibration of frequency
(o f and transient vibrations with the natural frequencies of the system.
We are interested only in the former. Let the steadystate vibration be
q. = //<V  q = \ e i°>ft
By putting this into (44) and canceling out e lC0ft , we have
( m t *a„ + c«)/,= gi  (<A+C)l = g (45)
This is a set of linear simultaneous equations that can be solved by standard
methods. If m f is equal to any one of the natural frequencies, resonance
takes place. If f is a periodic force, it can be expanded into a Fourier
series, and the response to f can be obtained by the method of superposition
similar to that described in Art. 1.8.
(C) FORCED VIBRATION — GENERAL SOLUTION
For the general solution of a forced vibration problem let us begin with
Aq + Cq = f
(3.7) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
Transforming it into the principal coordinates, we have
ARp+CRp=f
Premultiply both sides by R and utilize (34) to obtain
Mp + Kp = Rf
187
or
p + W 2 p = M i R f
(46)
(47)
With M a diagonal matrix, M 1 is easy to compute since it is also a
diagonal matrix with elements m^ 1 = l/ra n , m 22 = l/^ 2 2> . . . , m~^ —
\jm nn . It is also easy to show that (47) is the same as (27) by deducing
from (34) that
M!R = (A R) 1
Returning now to (46) and writing it in longhand form, we have
™nPi + KiPi = hifi + r 21 f 2 + * ' ' + r nl f n = j\(t)
m 22 p 2 + k 22 p 2 = r 12 f t + r 22 f 2 + h r n2 f n = j 2 (t)
m nnPn H" ^nnPn = r lnfl + r 2nj2 + * ' + f n njn = JnV)
These equations can be solved separately. For instance, according to (76)
in Art. 1.10, if the system is initially quiescent, the solution for p x is
MO
Vm u k n J
j\(t) sin co^t — t) dr
where
AW = 'li/iO) + r 21 f 2 (r) + • * ' + r nl f n (r)
Let us define a matrix
sin W/ = W/  i W 3 / 3 + ~ W 5 r 5 • • •
(48)
(49)
(50)
Since W is a diagonal matrix, the series above also represents a diagonal
matrix, which is
sin co x t •••
sin Y/t =
sin co 2 t
sin co J
188 THEORY OF MECHANICAL VIBRATION (3.8)
The solution to (46) or (47) can thus be written
p(f) = J (M K)"* sin W(f  r)j(r) rfr
Jo
= (M K)"* J sin W(f  t)R f(r) rfr (51)
if p(0) = 0andp(0) = 0.
The solution for q is therefore
q(0 = R p (0 = R(M K)y* I sin W(f  t)R f(r) Jr (52)
Jo
if q(0) = and q(0) = 0.
On the other hand, if the initial values do not vanish, the following
terms must be added to (51) and (52), respectively:
cos V/t p(0) + W 1 sin V/t p(0)
and
R cos V/t R 1 q(0) + R W 1 sin Wr R 1 q(0)
The reader is left with the proof of this statement. (See Exercise 3.17.)
3.8 Free and Forced Vibrations of a Damped System
(a) free vibrations
When there are viscous forces acting on certain parts of the system,
there is an energy dissipation. It is not difficult to see that if the forces
are proportional to the velocities or relative velocities at certain parts of
the system the rate of local energy dissipation is proportional to the
square of the velocity of the parts affected by the damping force. This is
true whether the damping forces are external or internal. According to
(7) and (8), the squares of the velocities are quadratic functions of the
generalized velocities q's. Hence the rate of energy dissipation for the
entire system can be written as
2F=b ii q i q i  2F = q B q (53)
The energy equation for the system is then 9
f + V + 2F =
Following the same argument used before, we arrive at
*<Ai + Mi + Wi = ° I Aq + Bq + Cq = (55)
9 Fis a scalar function called Rayleigh" s function. The factor 2 is included to give F
the same appearance as T and V and also to give certain symmetry to Lagrange's
equation, modified by Lord Rayleigh.
(3.8) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 189
Let the solution be of the type
ft = he"'  q = \e°>
We have
(j"fl w + sb {j + c w )/, =  (s 2 A + sB + C)l = (56)
For nontrivial solutions of (56) the determinant
D(s) = \s 2 aij + sb ti + c w  = (57)
This is an algebraic equation in s of 2«th degree. It has In roots, which
may be real or complex. The rest of the procedure is similar to that
discussed in Art. 2.7. The results can be stated as follows.
To each solution of (57), say s = s {i) , there is a vector \ (i) such that
q = <V^
is a solution of (55). There are In such solutions, and the general solution
of (55) is therefore
q = f I'V^ (58)
If s {i) is real, \ {i) has real elements. If it is complex, it must come as one
of a complex conjugate pair, and the vector I's associated with this pair
also form a complex conjugate pair. The real I's are determined only to
within an arbitrary real multiplication factor, and the complex I's are
determined only to within an arbitrary complex multiplication factor.
In any event, there are In integration constants to be determined by the
In initial conditions.
(B) FORCED VIBRATIONS
Forced vibrations of damped systems are described by a set of equations
having the general form
or (59)
A q + B q + C q = f(r)
(i) Steadystate response to harmonic forces and transfer functions. If
f(t) is a sinusoidal function of time
fi = g/ Mft I f = & iWft
the steadystate solution of (59) can be obtained by assuming that it has
the form
q. = l/^  q = \e iu >f l (60)
190 THEORY OF MECHANICAL VIBRATION (3.8)
This results in a set of linear simultaneous equations represented by
(_ W/ 2 A + /ft)fB + C )l = g (61)
Let us define a matrix function
D(ioj f ) = co f 2 A + ico f B + C (62)
The solution to (61) is then
I = D\ko f )g (63)
provided that the determinant
D(/<)^0
Otherwise steadystate solution is not possible, for the amplitude tends to
grow without limit, and resonance is said to exist.
A problem of particular interest is that in which all but one of the g's
vanish; that is
gi = g% = ' ' ' = gil = gj+l = * * = gn =
and
gi = 1 (64)
In other words, g is equal to the unit vector of they'th coordinate. Let us
call the amplitude of the response of a generic coordinate q t to this unit
harmonic force the transfer function t i} . Evidently, the different transfer
functions between the different pairs of coordinates can form a matrix.
The elements of this matrix are obtained by substituting (64) into (63),
and it is not difficult to see that this matrix is the inverse of the matrix D.
To extend the domain of definition of the functions into the complex plane,
we replace the argument ico f by the complex variable s. Hence
T(s) = D~\s) = (5 2 A + *B + C) 1 (65)
This transfer function matrix is useful in studying the transient response
of a damped system.
If the forcing functions represented by f are periodic, we can expand
them into Fourier series and then apply the principle of superposition to
obtain the steadystate solution in the usual way.
(ii) Transient response — indicial response and impulse response. The
transient response of a damped system can be investigated through its
indicial response and impulse response, as discussed in Art. 1.11. Similar
to the situation with the transfer functions, for systems with many degrees
of freedom there are many indicial response functions that can be defined.
Let us use the symbol i$\t) to denote the indicial response of a generic
(3.8) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 191
coordinate q i to a unit step force applied to theyth coordinate. In other
words,
ft = «P(0 I q = "<'>(*)
satisfies (59) under the conditions
/i/»" =/*i =/m = •••=/„ =
and (66)
fi = J"(0
together with the initial conditions
?,(0) =
WO) = o
q(0) =
q(0) =
(67)
When the indicial response functions are continuous for t > and
differentiable for / > 0, the associated impulse response function can be
obtained by differentiation with respect to t.
at
h< J ')(0 = T u(i) (0
at
According to the discussions in Art. 1.11, the general solution to (59)
under the initial conditions (67) can now be constructed from the impulse
response functions:
q(0 = W)h H, (/  t) +/ 2 (r)h'»(«  t) ■ ■ /„(r)h<">( (  r)] dr
Jo
JO j = l
To simplify this expression, let us form two square matrices U and H
whose columns are the vectors u's and h's, respectively; that is
»« = ^ h ti = up
and
The solution to (59) can then be written
qlt) = fhtfi  t)//t) rfr
« ^>
q(0= H(/r)f(r)^T
Jo
The impulse response matrix H can be obtained by differentiating U,
the indicial response matrix, which is formed by the solutions to (59) under
192
THEORY OF MECHANICAL VIBRATION
(3.9)
the conditions (66) and (67). The matrix H can also be obtained by
integrating the transferfunction matrix T(s) with the integration factor
e st , as described in Art. 1.11(d).
d 1 f c+ix
H(0 = U(0 = — T(s)e«ds (69)
at Itti J c i oo
in which c is any positive constant.
It is to be noted that in analyses of the vibrations of damped systems we
do not try to orthogonalize the equations by a suitable linear transfor
mation of the coordinates, as we do with undamped systems. There is an
orthogonalization procedure, 10 but it is seldom utilized because of compu
tational difficulties.
3.9 Semidefinite Systems
Most vibratory systems are connected with a stationary frame; as a
matter of fact this stationary frame may be considered a part of the system.
For such systems the equilibrium configuration is stable, and any variation
in the generalized coordinates must be accompanied by a change in the
potential energy of the systems. These systems are called positivedefinite
because their potential energy expression
V = i q C q > (70)
except at the equilibrium configuration
q =
On the other hand, a system is called semidefinite if it is possible for V
to vanish without having all the q\ vanish simultaneously. la other words,
q = corresponds to a condition of neutral equilibrium.
Mathematically, the necessary and sufficient condition for the potential
energy expression to be nonnegative is that
c w c \i c \z
•ii >o
11
12
c 12
C 22
>0
C 12
C 2 2
C 23
C 23
C 33
>0
ICl =
>0
10 See "Coordinates Which Uncouple the Equations of Motion of Damped Linear
Dynamic Systems" by K. A. Foss, Journal of Applied Mechanics, Vol. 25 (1958),
pp. 361364.
(3.9)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
193
For positivedefinite systems all of the determinants are greater than zero
and for semidefinite systems the last one or more of the determinants are
equal to zero. In vibration study it is seldom necessary to evaluate these
determinants, since an examination of the makeup of the system is usually
enough to determine whether or not it is semidefinite. If the system does
not include a stationary frame, or the stationary frame is not positively
connected to some part of the system, the system is semidefinite. 11 Two
such systems are illustrated in Fig. 70.
AAA/
SS
S3J
WW
(a)
_J
vww
mmm
S3J
Note that we are not concerned about the kinetic energy expression,
since any motion of a system must result in a positive amount of kinetic
energy.
The analysis presented so far in this chapter is essentially valid for
semidefinite as well as positivedefinite systems, but there are certain
features of semidefinite systems that we shall discuss.
Since for a semidefinite system
M = o
one or more vectors I exist, so that
c»h =
IC! =
CI
(71)
(72)
This means that (15) can be satisfied with to = 0, and (13) can be satisfied
by
/,(** + p)
q = Kaf + p)
(73)
11 We assume that the equilibrium configuration is at least not unstable so that the
potential energy is nonnegative.
194 THEORY OF MECHANICAL VIBRATION (3.9)
which represents motions having constant velocities. Semidefinite systems
are thus associated with zero roots of the frequency equation (16). The
number of independent nonzero I's which satisfies (72) is the same as the
multiplicity of the root co 2 = in (16). Together, they represent some
of the natural modes of the system, which may be called the zero modes.
In mechanics there are two types of problems involving the study of
semidefinite systems. In the first type we are given systems known to be
semidefinite, systems such as a rotating shaft, a moving vehicle, or a
rocket in free flight. We are to analyze their vibrational modes in addition
to the known zero modes. In the second type of problem the potential
energy expression V or the elastic matrix C includes some parameters
that must be determined so that the system can be made semidefinite.
This is the type of problem studied in theory of elastic stability.
For a positivedefinite system the elements in the inverse of its elastic
matrix, C 1 , are known to be the influence coefficients. For a semidefinite
system the matrix C is singular because of (71), so that its inverse is not
defined. This agrees with the physical facts that when a system is in neutral
equilibrium its influence coefficients are mostly undefined quantities.
(See Exercise 3.15.) This has two practical consequences. The first is
that we cannot write the alternate form of (23)
[Ci A]r = 1 r
CO*
which is a desirable form for some methods of numerical computation,
discussed in Art. 3.11. The second is that without the use of influence
coefficients it becomes inconvenient to set up the equations of motion for
some systems, such as a beam carrying three masses in free flight.
The zero modes of a semidefinite system are not of interest in vibration
study. It is possible to "suppress" these modes by introducing constraints,
which will reduce the number of degrees of freedom and in the meantime
convert the system into a positivedefinite one. This procedure is illus
trated in Art. 2.9(a) and may be generalized as follows. Let 1° represent
a zero mode. Then
Cl° =
Since C is symmetrical,
i°c = o
Premultiply (13) by 1° to obtain
T°Aq + !°Cq =
or
1° A q = i°Aq = af + />
(3.9)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
195
Suppressing the zero mode is equivalent to letting a = and b = 0.
In other words, only those motions orthogonal to the zero mode are
considered. This results in a constraint relation between the g's. Written
longhand, the relation is
('All + / 2 %j + • • • + />m)?l + (/xV + / 2 °«22 + ' * ' + '>2n)?2
+ ' ' • + (h°a ln + l 2 «a 2n + ' ' • + l n «a nn )q n = (74)
For each zero mode there is a constraint relation such as the one above.
With these constraints, a corresponding number of the coordinate variables
J T
<?2
TTI3
Figure 71
g's can be eliminated. This elimination process results in a positive
definite system having fewer degrees of freedom. 12
It is not necessary that the constraint which suppresses the zero mode
be obtained by finding nontrivial solutions of (72). In many cases it can
be obtained by simple principles of mechanics. As an example, let us
consider the free vibration of a light rectangular beam carrying three
particle masses, as shown in Fig. 71. The beam is unsupported in free
space. This vibration is often called the free freevibration. Obviously,
the degrees of freedom consist of three translations, three rotations, and
two vibrations in the two principal planes of the beam cross section. Of
the eight degrees of freedom, only two are nonzero modes. To solve this
problem, we can suppress all the zero modes by assuming that the system
has no linear momentum and no angular momentum. Furthermore, we
restrict the motion in one of the two principal planes of bending of the
beam. This is permissible, since obviously there is no coupling between
the two vibrations in the two planes of bending. Referring to Fig. 71, we
have therefore
mtfi + m 2 q 2 + m 3 q 3 =
m#! • (2a) + m 2 q 2 • (a) =
For an illustration of a similar problem see Art. 3.11(e).
196 THEORY OF MECHANICAL VIBRATION (3.10)
Upon integration and setting the integration constants equal to zero,
we have
Hence
T= 2 ( m &* + m2 ^ 2 + m ^
1 / _ 4m] 2
h
1 1 48 F/
K= 2 W2 = 2W [K?1+?3)  ?J!
2 \ m<> mo/
3^
.2 \ mo/ 777o .
2
?! 2
where i^ represents the bending stiffness of the beam cross section. For
the vibration in the other plane of bending we need only change I ± to L 2
in the potential energy expression.
3.10 Repeated Roots of the Frequency Equation
In Art. 3.5 it was shown that if the roots of the frequency equation
(16)
C  w 2 A =
are distinct there is a modal vector to each of these roots. These modal
vectors are orthogonal to one another, with respect to both A and C.
po) A r<« =
CO; ^ (Oj
pw) c r<*> =
The orthogonal relationship among the modal vectors also insures that
they are linearly independent.
Now if some of the roots of the frequency equation repeat themselves,
we will have fewer distinct natural frequencies. The question is then
"Does the system have fewer natural modes?" The simple examples
shown in Arts. 2.10 and 2.14(a) indicate that the answer is negative.
A rigorous mathematical treatment of the problem is not necessary for
our purpose. We simply state that in the eigenvalue problem (23) arising
(3.11) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 197
from freevibration analysis the coalescence of eigenvalues does not cause
a coalescence of eigenvectors. This statement, however, is not generally
true for an arbitrary eigenvalue problem.
Lr = 2r
To summarize, let
be an Mime repeated root of the frequency equation (16). There are r
linearly independent modal vectors, r (1) , r (2) , • • • , r (r) , representing
different modes of free vibrations having the same frequency. These
modal vectors are orthogonal, with respect to A and C, to all other modal
vectors of the system not belonging to this set. Among themselves the
orthogonality relationship may or may not hold. However, the set of
modal vectors corresponding to a given repeated root are not uniquely
determined because any linear combination of these vectors can also
serve as a modal vector, since
(A 1 CXajH" + a 2 r< 2 > + • • • + a r r (r) )
= co r 2 (air (1) + a 2 r< 2 > + • • • + a r r<'>)
It now becomes possible to construct a new set of modal vectors by taking
different linear combinations of the old set, so that the vectors in the new
set are orthogonal among themselves with respect to A and C. The method
of construction is left to the reader as an exercise. (See Exercise 3.16.)
SECTION B. METHODS AND APPLICATIONS
3.11 Solution of Eigenvalue Problems by Matrix Iteration
The direct method of solving the eigenvalue problem
(0 2 A r = C r
as outlined in Art. 3.4, is not practical when a large number of coordinates
is involved because of the excessive amount of arithmetic computations
needed. There are several numerical procedures that simplify computa
tional steps in one way or another. Aside from the amount of labor
involved, the question of checks, control of roundoff errors, susceptibility
to mistakes, etc., must also be considered. Complete solution of large
scale eigenvalue problems is practicable only through the use of electronic
digital computers. Numerical procedures, which are suited for program
ming on such computers, have requirements of their own, but this is a
subject in itself and cannot be dealt with exhaustively here. In this article
198 THEORY OF MECHANICAL VIBRATION (3.11)
we can discuss only a few methods that may be used to solve problems of
reasonable size with the help of a desk calculator and slide rule.
(a) determination of largest eigenvalue by iteration
The basic principle of the matrix iteration procedure is extremely
simple.
Let the eigenvalues of a matrix L be
?i 1 , X 2 , •  • , ). n
with
\K\ > N > \h\ ■ ■ ■ \K\
Designate the corresponding eigenvectors by
r (D r (2) . . . r (n)
Let v be an arbitrary vector. It can be thought of as a linear combination
of the eigenvectors, since the eigenvectors are linearly independent.
v = ai r (1) + a 2 r (2) + • • • + y. n r {n)
Hence
Lv = a 1 L r<« + a 2 L r< 2 > + • • • + a B L r<»'
= OjL^rW + a^,r«» + ■ • • + a n A„ r<»'
By repeating this process, we have
L 2 v=LLv = a 1 A 1 2 r<D + a 2 A 2 2 r< 2 > + • • • + a n A B 2 r<»)
L* v = a 1 A 1 V (1) + a 2 A 2 l > (2) + • ■ * + «„;„><*)
Since
W > 1^1 > • • • 4
as / becomes sufficiently large, the first term on the righthand side
predominates over the rest of the terms and
Thus U v is parallel to the eigenvector r (1) , and the corresponding eigen
value X x can be obtained by operating it once more by the matrix L.
We thus have a method of determining the largest (in absolute value)
eigenvalue and the associated eigenvector of a matrix L. The method
consists of repeated multiplication of an arbitrary vector by the matrix L.
The rapidity with which the eigenvector r (1) is approached depends upon
two things: the closer the starting vector v parallels r (1) , the faster the
convergence. In other words, it is desirable that the coefficient x 1 be
(3.11)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
199
larger than the rest of the a's. Theoretically, there is a possibility that by
accident a starting vector with a x = will be chosen. Then the iteration
process described will converge to the eigenvector with the next largest
eigenvalue. Practically, this possibility does not exist, for even if a x =
in v, unavoidable roundoff errors will introduce a small component
along r (1) in the iterated vector, and this component will eventually
dominate the rest. The second thing that enhances the rapidity of con
vergence is to have the eigenvalue X x much larger in absolute value than
the rest. This is especially important in large matrices.
Let us now illustrate the procedure by a simple example. One small
modification will be made in the illustration. To keep the numerical value
of the elements of the iterated vectors within a certain range, they are
normalized after each iteration. A selfexplanatory tabulation scheme is
also used.
L
V
Lv
v'
Lv'
v"
Lv
y«T
5
0.6
1
5.6
1.0
6.74
1.00
7.79
1.00
2
4
1.2
1
7.2
1.3
10.68
1.86
15.02
1.93
7
4
5
I
16.0
2.9
26.7
4.65
37.69
4.85
L
V
Lv
yiV
L v iv
v v
Lv v
A
5
0.6
1.00
7.91
1.00
7.96
1.00
7.98
(7.98)
2
4
1.2
1.93
15.54
1.96
15.76
1.98
15.86
(8.01)
7
4
5
4.85
38.97
4.93
39.45
4.96
39.72
(8.01)
Thus the approximate answer is
X x = (7.98 + 8.01 + 8.01)/3 = 8.00
pa) = [1.00 1.98 4.96]
The exact answer of this problem happens to be
l x = 8.00
F< 1} = [1.00 2.00 5.00]
The procedure illustrated is straightforward. Such routine calculation
can be put into the hands of untrained office help. On the other hand, if a
little ingenuity is exercised, the process can be speeded up. For instance,
we must realize that each iterated vector is a trial vector for the next step,
200
THEORY OF MECHANICAL VIBRATION
(3.11)
so that it may be changed at will. After a few steps, a trend of changes in
the elements is discernible. The trial vector of the succeeding step can be
altered judiciously from that given by the previous step to hasten the
convergence.
The matrix L used in this example was obtained from the following
problem :
T = (21<7i 2 + 19<? 2 2 + 5q z 2 + 14^ 2 + \4q 2 q 3 + 2q Y q z )
V = i(14^ 2 + 6? 2 2 + 2q 2 + 6 Ma + 2q 2 q 3  4 Ma ) x 10 3
x 10 3
A 1 C= 125 x L
The reader will profit by solving this eigenvalue problem by the classical
method. The answer is
21
7
r
r l4
3
1
7
19
7
C =
3
6
1
1
7
5
2
1
2
(0 ± 2 =
1000
r n
r (i) =
2
5
500
r<« =
1
■4/3
5/3
r^ 6 ' ^
(b) determination of the lowest natural frequency
In the majority of vibration problems it is the lowest natural frequency
that is of interest, whereas the method illustrated gives only the highest
eigenvalue. This situation can easily be remedied if our eigenvalue problem
is formulated as
(Ci A) r =  r
CO 1
The largest eigenvalue of C _1 A will then correspond to the lowest natural
frequency. In this formulation the system must be positivedefinite so
that C _1 is defined. 13
Example: Given
T = «3& + 4^ 2 2 + 2^ 3 2 + q, 2 + 4q,q 2 + 2^ 3 + 2q 3 q A ] X k t
V = M4?i 2 + 5</ 2 2 + q 3 2 + 2? 4 2  4q x q 2  2q 2 q 3 ] X k v
in which k t and k v are constants having appropriate units.
13 For iteration of a semidefinite system see "Iteration in Semidefinite Eigenvalue
Problems" by B. M. F. de Veubeke, Journal of Aeronautical Sciences, Vol. 22, No. 10,
October 1955. Also, many physical problems are formulated by influence coefficients
to begin with. The influence coefficients form the matrix C _1 and can be used directly
in the ensuing computation.
(3.11) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 201
To find the lowest natural frequency and the associated modal vector
A =
3
2
01
2
4
1
1
2
1
1
1_
x k t
4
2
2
5
1
01
1
1
2
x k,
C! =
8
4
4
01
4
8
8
4
8
32
12
24/c,
2
1
1
01
1
2
2
1
2
8
3
6k
2
1
1
01
1
1
2
2
2
8
X
3^
C!A
The iteration of the last matrix:
3
2
0^
2
4
1
1
2
1
1
1
8
9
3
11
7
12
6
2
7
18
18
8
3
3
±1
6A:„
L
8
9 3
7
12 6
7
18 18
3
1
21 1.0
27 1.0
1
27 1.3
38 1.4
1
51 2.5
79 2.9
1
6 0.3
8 0.3
b
c
d
e
29.6
1.00
29.80
1.000
41.8
1.41
42.12
1.412
86.8
2.93
87.68
2.935
9.6
0.32
9.75
0.327
A note about another scheme that often simplifies computation. We
observe that in the foregoing tabulation
Hence
Or
L a = b and L c
L (c  a) = d  b
d = b + L (c  a)
Since (c — a) is small, d can be conveniently obtained by applying a smal
correction to b. Similarly, the next step is 14
f = d + L (e  c)
"29.80"
42.12
87.68
9.75
+
9
12
18
3 11
r o n
"29.841
6 2
0.002
42.19
8 8
0.005
87.86
3 3
0.007
9.78
14 This correction can be carried out within the regular tabulation without writing
out the numerical equation shown here.
202
THEORY OF MECHANICAL VIBRATION
(3.11)
As a final check, let us get the ratio between the corresponding elements of
f and e.
29.84
1.000
87.86
2.935
= 29.84
= 29.93
42.19
1.412
9.78
0.327
= 29.88
= 29.91
Thus the lowest natural frequency of the system is
< = ^x L = 0.200^
1 k t 29.9 k t
and the amplitude ratio of this mode is
1 : 1.412 : 2.935 : 0.327
(C) USE OF RAYLEIGH'S QUOTIENT
If we are interested only in the natural frequency, we can stop the
iteration process sooner and use Rayleigh's quotient. Suppose we stop
at the vector a.
a=[l 1.4 2.9 0.3]
Rayleigh's quotient can be computed by the following tabulation:
c
a C a
" 4
2
0"
1.0 x 1.2 = 1.20
2
5
1
1.4 x 2.1 = 2.94
1
1
2.9 x 1.5 = 4.35
2_
0.3 x 0.6 = 0.18
I C a = 8.67 x k
A
a A a
"3
2
0"
1.0 x 5.8 = 5.80
2
4 1
1.4 x 10.5 = 14.70
z t
1 2
1
2.9 x 7.5 = 21.75
1
1_
3 x 4.6 = 0.96
a A a = 43.2 x k t
< =
8.6
A3.
IK
6k t
= 0.201 ^
(d) determination of a second eigenvector
To get other eigenvectors by iteration, we need some supplementary
theorems. These are now discussed.
(3.11) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 203
Let the eigenvalues and the eigenvectors of a matrix L be
a>i, A 2 , • • • , A n
and
(1) (2) . . . r (n)
The matrices al_ and L — pi, in which I is the identity matrix, have the
same eigenvectors and their eigenvalues, respectively, are
aA l9 aA 2 , • • • , <xX n
a ±  & a 2  n, • • • , (x n  p)
The proof of this theorem is simply as follows:
(aL) r<*> = a [L r«>] = oV™
(L  pi) p«) = L r«>  j8r<« = (A,  0)r«>
Next, let L and M be two matrices having the same set of eigenvectors.
Let their eigenvalues be, respectively,
and
X l5 / 2 ) '
' 5 *n
The product matrix P,
/"l, ^ 2 » • •
* , /«,
P = L M
also has the same set of eigenvectors with eigenvalues
^l/*l> *2^2» ' ' ' > ^n^n
This theorem is true because
P r<<> = L(M r (l '>) = L^r<« = A^r<*>
From these two theorems we can deduce that the matrix
P = L(^I  L)
has the same eigenvectors as L, and its eigenvalues are
0> A 2 (^i — ^2)5 ^3(^1 ~~ ^3)' " * " 5 ^nWi — A n )
This result gives us a way of evaluating eigenvalues, other than the largest
one, by the same iteration process described.
Let us assume that we are given a matrix L whose eigenvalues are known
to be real positive numbers :
X x > A 2 > A 3 • • • > X n >
First find A x and r (1) by iteration. Afterwards, form the matrix
P = LtfJE  L)
which has the same set of eigenvectors as L. Then iterate P to get the
eigenvector with the largest eigenvalue. This eigenvector cannot be the
r (1) just found, since the eigenvalue of P corresponding to r (1) is zero
204
THEORY OF MECHANICAL VIBRATION
(3.11)
and therefore cannot be the largest one. To which of the eigenvectors the
iteration of P will converge cannot always be established beforehand.
We have, however, a useful clue. If
k > a, + a 3 )
it is easy to show that
* A  h) > UK  h) > UK h)
In that case the iteration of P will yield the eigenvalueeigenvector pair
UK ~ h) and r< 2 >
We can also show by theory of algebraic equations that the sum of the
diagonal elements of L is equal to the sum of its eigenvalues. Or
'n + h% + • • ■ + l nn = K + A 2 + A 3 + • •  + K
Hence, knowing A l5 we know also the value of A 2 + A 3 + . . . + X n .
Very often this sum is less than X x and the inequality
/Ij !> /l 2 4* a 3
is assured.
Let us now return to our previous example. There we have found the
largest eigenvalue A x of the matrix L.
"8 9 3 r
7 12 6 2
7 18 18 8
3 3
A x = 30
The sum of all four eigenvalues of L is 8+ 12+ 18 + 3 = 41. Hence
A x is larger than the rest of the eigenvalues combined, and the largest
eigenvalue of the matrix 15
P = L(A X I  L)
is
X 2 (X X — A 2 )
"8 9 3 1"
"(308) 9 3 1
p =
7 12 6 2
7 18 18 8
_0 3 3_
X
7 (30  12) 6 2
7 18 (3018) 8
03 (30  3)
" 92 36 45 23"
28 45 27 25
98 63 63 29
21 54 27 57_
15 It i
5 not necessary to 1
ise tl
ie exact value of X x in the p
rocedure that follows.
(3.11) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
Iteration of this matrix yields
205
and
or
7< 2) = [1.000 0.565 1.261 0.670]
A 2 (30  A 2 ) = 184.5
A 2 2  30A 2 + 184.5 =0
30  V900  738
8.62
The other root of the quadratic equation has no meaning. To get accuracy
as well as a check, we should go back to the original matrix and compute
L r< 2 >.
8
9
3
1
1.000
8.634
1.000
7
12
6
2
0.565
4.874
0.565
7
18
18
8
1.261
10.882
1.261
3
3
0.670
5.793
0.671
The answer checks out, and the eigenvalue is A 2 = 8.63.
To get another eigenvector, we can form the matrix Q,
Q = L(30I  L)(8.63I  L)
and repeat the process as before.
We conclude this discussion with two remarks.
(i) Although theoretically by iterating the matrix (^1 — L) we should
get the eigenvector r (7l) having the smallest eigenvalue, it is usually not
practical to do so because the first few eigenvalues of this matrix {X 1 — X n ),
(A x — A 7 j_ 1 ), etc., are often close together so that convergence is very slow.
(ii) The multiplication of two n x n square matrices represents as much
computational labor as multiplying a square matrix to a column matrix
n times. Therefore, for moderately large matrix iteration problems labor
may be saved by not getting the product matrix P (or Q) first. Instead,
we can iterate a trial vector by multiplying it alternately with (XJ. — L)
and L.
It is not even necessary to iterate with these two matrices alternately.
One can be used more often than the other. The principle is as follows.
Let the starting vector be
■ d)
v = ai r u > + a 2 r^> \ 1 a n r
(2)
,(n)
206 THEORY OF MECHANICAL VIBRATION (3.11)
After being operated on by (XJ. — L), the result is
v' = 6 r<» + a^  A 2 )r< 2 > + • • ■ + *„Oi  A B )r<«>
Theoretically, e is zero, but because of unavoidable roundoff errors as
well as the error in the value of X x , v' still retains a small component in
r (1) . Next, we operate v' with L i times:
V v' = 6V (1) + * 2 V(*i  h)r (2) + • • • + *nKKK  K)r {n)
By this time the coefficient of r (1) , that is eXf, may become a fair amount.
So we suppress it by operating again with (^1 — L) and then follow
by operations with L.
The sequence of operation is thus
V(X x l  L) L V^l  L) v
It is clear that the greater the precision in the value of X 1 previously obtained
and the smaller the roundoff errors the less often it is necessary to use the
matrix (A X I — L). It is also true that if additional eigenvalues are to be
found, regardless of the scheme adopted, the preceding eigenvalues must
be determined with greater accuracy. The last remark applies also to the
method discussed below.
(e) determination of successive eigenvalues by successive
reduction of the order of the problem
When one of the modes of vibration is known, it is possible to reduce
the order of the problem by one. The situation is exactly the same as that
discussed in Art. 3.9, in which a semidefinite system is reduced to a positive
definite system by introducing the constraint relation (74) to suppress the
zero mode. The procedure is best illustrated by an example. Take the
eigenvalue problem we have just studied. After we have found the first
eigenvector,
H 1 * = [1.000 1.412 2.935 0.327]
we introduce either of the constraint relations
? C r<« = and FA r™ =
This is equivalent to restricting the vibrations only in the remaining
modes, which are all orthogonal to r (1) with respect to either C or A.
Putting in numerical values, we have for this example
(4 x 1.000  2 x 1.412>i + (2 x 1.000 + 5x 1.412  1 x 2.935)r 2
+ (1 x 1.412 + 1 x 2.935)r 3 + (2 x 0.327)r 4 =
or
r t = 1.810r 2  1.301/<3  0.531r 4
(3.1
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
207
By substituting this relation into the original problem,
8
9
3 1"
>r
>r
7
7
12
18
6 2
18 8
'3
= A
'2
3 3 _
_^4_
/4.
we will have four linear equations with three unknowns. One of these
equations can be considered redundant and be left out. If we leave out
the first of the equations, the following reduced system is obtained:
^12  (7 x 1.810) 6  (7 x 1.301) 2  (7 x 0.531)'
18 (7x1.810) 18 (7x1.301) 8  (7 x 0.531)
 (0 x 1.810) 3  (0 x 1.301) 3  (0 x 0.531)
This procedure can be generalized as follows. Find a constraint relation
through the orthogonality property of eigenvectors and express it as
r x = a 2 r 2 + a 3 r 3 + • • • + a n r n
The problem is then to find the eigenvalues of the reduced matrix L':
'22 H" '21 a 2 '23 + '21 a 3 ' ' ' '2n H~ '21 a n
r ^ 2 ] r^ 2 "
r 3 = 1 r 3
_ r 4_ L r 4_
L' =
'32 i '31 a 2 'as ~r «ai a
32
! 31^2
33
31^3
'zn i '31 a «
J.i
+ /»1«2 / n 3 + 4l«S
'nrc T" 'nl*r!,
The eigenvectors of L' have only (n — 1) elements, which are the same as
the corresponding elements of (n — 1) eigenvectors of L. The remaining
elements of the eigenvectors of L are obtained by the constraint relation
imposed.
(F) REMARKS
So far, in this article and in the next, the discussion centers on the
mathematical problem of determining the eigenvalues and eigenvectors of
a square matrix. To connect this discussion with the solution of physical
problems, the following remarks are presented to bring together pertinent
results previously obtained.
Assume that we are dealing with vibrations of systems in which damping
forces can be neglected. For such systems the first thing of interest is the
natural frequencies and modes of vibration. This we have shown to be an
eigenvalue problem of the square matrix A 1 C or C^ 1 A. Since the
numerical methods discussed always give the largest eigenvalue first
and since in practice we are more interested in the lower frequencies, the
matrix product C _1 A (hence C 1 ) has to be found. The matrix C 1 is
208 THEORY OF MECHANICAL VIBRATION (3.12j
formed by the influence coefficients between different pairs of generalized
coordinates. If these coefficients can be computed directly, a tedious and
errorproducing inversion procedure can be eliminated. For systems
such as beams and frames, carrying masses, the influence coefficients are
directly obtainable from formulas in strength of material, whereas the
elastic constants, which form the elastic matrix C, are not. On the other
hand, there are systems whose potential energy expressions in terms of
generalized coordinates can be readily obtained. For such systems it is
more convenient to determine C first and then invert it to obtain C _1 .
After the matrix product C 1 A is obtained, the methods described can
be used to determine the natural frequencies and amplitude ratios of the
system in free vibration. Ordinarily, only the first few modes are of
practical interest. This is because of two facts. First, the frequencies of
external disturbances are usually far below the higher natural frequencies.
Second, the assumptions required in setting up the differential equations
are often very unrealistic for the higher modes of vibration, although they
may be justified for the lower modes.
When the numerical procedure is terminated after the determination of,
say, the first k modes, only the first k columns of the modal matrix R
defined by (18) are known. By knowing the first k columns of R. we can
compute the first k diagonal elements of the matrices M and K, since
M = R A R and K=RCR
In this way the first k equations represented by (46) can be obtained. By
solving these equations for p 1 ,p 2 , • • • , p k anc * then assuming/?^ = p k _ 2 —
. . . = p n = 0, we can get an approximate solution for q l9 ql, . . . ,q n in a
forced vibration problem. This procedure is permissible when the fre
quency of the external force or forces is low in comparison with the higher
natural frequencies of the system. For example, a system having 7 degrees
of freedom is under an external sinusoidal force of 60 cycles per second.
The first three natural frequencies were found to be 44, 185, and 602
cycles per second. The system's approximate response to this external
force can then be determined without finding all of its natural modes.
3.12 Additional Theorems and Methods
(a) enclosure theorem
In many practical problems the generalized coordinates of the system
are the displacements of the different inertia elements with respect to some
fixed reference system. For such problems the kinetic energy expression
contains only the squares of ^'s.
T = Jfaitfi 2 + «22^ 2 2 + * * * + a nn q n 2 )
(3.12) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 209
In other words, the matrix A is a diagonal with positive diagonal
elements.
a {j = when i ^ j and a {j > when i = j
In matrix iteration of the eigenvalue problem for such a system we have
the socalled enclosure theorem, which can be stated as follows. Let
L = A 1 C
and
v' = Lv
in which v is an arbitrary vector. Let the ratios between corresponding
elements in v' and v be
A l — > A 2 — > * ' ' j A n —
Vl v 2 V n
Let the largest of these ratios be A^ ax and the smallest, A f min . The theorem
then asserts that between these two limits there is at least one eigenvalue
of L. 16
The conditions stated for the validity of this theorem are rather stringent.
For example, it does not apply to the case
L v = (C 1 A) v = v'
in which A is diagonal and C _1 is symmetric. However, these conditions
are sufficient conditions. Although we have examples to show that the
theorem is not valid for arbitrary L, it is suspected that it is valid under
more liberal conditions, which at present are not known. Experience
seems to indicate that when A.' maLX and A' min are reasonably close together they
usually bracket an eigenvalue.
(b) eigenvalue problem of a symmetrical matrix and
symmetrization of a general eigenvalue problem
The eigenvalue problem of a symmetrical matrix
Sr = Ar (S = S)
has many convenient features. One is that the enclosure theorem just
discussed is valid; another is that Rayleigh's quotient of vectors during
an iteration process is easily computed :
vSv v v'
R(y) = =— = =
V V V V
16 See Eigenwertaufgaben mit technischer Anwendung, by L. Collatz, Akademische
Verlagsgesellschaft, Leipzig, 1949, p. 289.
210 THEORY OF MECHANICAL VIBRATION (3.12)
Still another is that the eigenvectors are mutually orthogonal in the
ordinary geometrical sense; that is
Among other things, this simplifies the procedure described in Art. 3.1 1(d).
Moreover, if the eigenvectors are normalized in length, that is, by proper
scaling of these vectors so that
f«> r a) = i
then the modal matrix formed by the r's will have the very desirable
property that
R R = I or R = R 1
The truth of these statements can be readily verified.
Now we want to show how a general free vibration problem can be
converted into that of a symmetrical matrix. The idea is simply to convert
one of the energy expressions into a sum of squares by a convenient linear
transformation of variables. Suppose we choose the potential energy
expression for this maneuver.
V = iq C q
First we find a matrix F such that
FF= C
This is always possible when C is symmetrical. Introduce a new set of
coordinates represented by g.
g=Fq or q = F 1 g
This transformation changes the energy expression into
K=iqCq = iqFFq = igg
T= iq Aq = ^(F 1 A F"i) g = Jg S g
where
S = F 1 A F 1
can be shown to be symmetrical. The eigenvalue problem then becomes
S r = Ar I = i ■
or
Similarly, if we make
FF = A and FSF=C
we have a symmetrical eigenvalue problem with X = co 2 .
At first glance it appears that in our effort to symmetrize the problem
(3.12)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
211
a great deal of additional labor of computation is introduced to defeat
our original purpose. But this is not so, for by a proper choice of F the
computational labor involved will be no more than that of getting C _1 A
or A 1 C, and in many ways it is more convenient to manage.
The matrix equation
FF= C
represents a system of n 2 linear simultaneous equations, of which n{n — l)/2
repeat themselves because C is symmetrical. Therefore, the elements of F
are not uniquely determined and n(n — l)/2 of the elements may take
arbitrarily assigned values. Since the object is to find as simple an F as
possible, we will assign zero as the value of these elements. The most
advantageous way of placing these zeros turns out to be all at one side of
the diagonal line. In other words, F is a triangular matrix.
711 712 713 * ' * fin
722 723 ' ' ' J2n
f„ ••• f 3n
F =
0— /„„.
The expansion of the matrix equation FF = C yields the following
simple relations:
Tii = c n
7ll/l2 == C 12' J11J13 = C 13' ' ' ' 'Jlljln = C \n
7 22 = C 22 — 712
722/23 = C 23 — 712713' 7 227 24 = C 24 — 712714'
7 33 == C 33 ~~ 713 ""7 23
7 33J34 = C 34 ~~ 713714 ~ 7 23724' 7 337 35 = C 35 — 7 137 15 ~~ 7 23725' '
f 2 = C — f 2 — f ''
J nn ^nn Jin Jin
■f)
The salient feature of these equations is that the elements of F can be
computed one at a time starting from the left to the right in each row and
then row after row. A convenient tabulating schedule can be set up to
handle this computation.
After having found F, we will now find F _1 . Again a triangular matrix
has the advantage that its inverse can be computed one element at a time
in a simple way. This is in marked contrast to the determination of the
inverse of an arbitrary matrix. As a matter of fact, there is a number of
practical schemes for the determination of the inverse of a matrix based
on the principle of splitting the matrix into two triangular matrices.
212 THEORY OF MECHANICAL VIBRATION (3.13)
Since f it = when i >/', the expansion of
F 1 F = I
is ./irVn = i.
fll~fl2 + fl2~ J22 =
/ll 7l3 +/l2 723 + /l3 7 33 = ^
/ll /in + /l2 72n + ' ' + fin fnn ~
/irVii = or / 21 i =
J22~~ J22 = 1» J22~J2Z + 7 23~ 7 33 =
/«r x = o, /„* = o, As" 1 =
/■ _1 /" =1
J nn J nn L
It is seen that F 1 is also a triangular matrix, and its elements can be
computed from F one at a time in a simple way.
3.13 Chain Systems — Holzer's Method
One way of obtaining the frequency equation of the vibration problem
embodied in
(Ca  a ii o^)r i = I (C  Aco 2 )r =
is as follows. Set aside one of the n simultaneous equations represented
by this matrix equation. Solve the remaining n — 1 equations by assigning
an arbitrary value to one of the coordinate variables. The solution will be
in terms of co 2 . By substituting this solution into the equation that has been
set aside, we obtain an equation in co 2 , which is the frequency equation of
the system. This method is usually too tedious. Under some special
circumstances, however, the principle of this method can be applied to
determine the natural modes.
Consider the torsional vibration of a shaft carrying a number of disks.
as shown in Fig. 72. The eigenvalue problem for this system is that of the
following set of algebraic equations. [See Art. 2.9(a).]
+ /jW 2 ^ = fc 18 (^  </> 2 )
^12(^1 — ^2) + h c ° 2< t ) 2 ~ ^23(^2 — ^3)
^23(^2  </> 3 ) + V^Vs = ^34(^3  4 )
KlJtn1 ~ <f>n) + />Vi =
(3.13)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
213
Physically these equations describe how the torque of the shaft changes
from one section to the next. We start from zero torque at the left and
end up with zero torque at the right end.
Holzer's method of solving this problem is as follows. Assume first
a value for co 2 and assign the value 1 to </> l5 which can be arbitrary anyhow.
From the first equation we then get ^ x — (/» 2 and (/> 2 . By putting these into
the second equation, we obtain </> 2 — </> 3 and </> 3 , and so forth. When we
reach the (n — l)th equation, we shall have all the values of the </>'s. The
last equation then tells us whether or not our assumed value for co 2 is
correct. By trial and error a value for co 2 will eventually be found that will
: 23
&34
i¥
Figure 72
make the lefthand side of the last equation vanish or become very small.
This value is the square of a natural frequency, and the corresponding
values of (/>!, <£ 2 , . . . , <j> n represent the elements of the corresponding modal
vector.
To facilitate the setting up of a tabular method for the computations,
the preceding set of equations are rewritten:
^co 2 ^ + I 2 w 2 <f> 2 = ^23(^2  <W
I^j 2 ^ + / 2 co 2 (/> 2 + I s co 2 (/> 3 = k u ((j> 3 
/ lQ >Vi + Vo 2 <£ 2 + • ■ • + K^ n =
#1)
The computation for each trial value of co 2 consists in filling out Table 1 .
Columns A and F are first computed from the given data with an
assumed value of co 2 . The rest of the places are filled in according to the
sequence indicated. Item D n represents the lefthand side of the last
equation, which is a function of the value of co 2 assumed beforehand.
The idea is to determine the value of co 2 for which
D n (co 2 ) =
To start the trialanderror procedure, we must have a reasonable estimate
of the correct co 2 , lest the labor needed be prohibitive. To this end we
214
THEORY OF MECHANICAL VIBRATION
(3.13;
first compute Rayleigh's quotient with a roughly estimated amplitude
ratio </> x : <f> 2 : . . . : </> n . Since the system is semidefinite with a zero mode, it
is necessary that this estimated ratio satisfies
h<h + IJt + • ' • + h<f>n =
Rayleigh's quotient obtained with this constraint will always be larger
than the square of the lowest nonzero natural frequency of the system.
It is therefore preferable to start our trialanderror procedure with a
value of co 2 somewhat lower than that of Rayleigh's quotient. 17
(1)
(2)
(3)
00
Table 1
CO 2 =
A
B
C
D
E
F
Im 2
4>
Ioj 2 (f>
^Ioj 2 cf>
= k \<f>
ZIoj 2 4>
\4
Hk
k ^
l x OJ 2
hco 2
I^ 2
I n OJ 2
1.000
B 1 E 1
B 2 — E 2
A 1 B 1
A 2 B 2
AB 3
c,
C 3 + D 2
D 1 F 1
D,F 2
D3F3
\Jk 12
1/^23
1/^34
After the first trial, it becomes necessary to have a way of knowing
which way to correct the co 2 value for the second trial. In other words,
we must know how D n varies with co 2 in the vicinity of the trial value.
It was mentioned previously that D n (co 2 ) = is the frequency equation 18
if co 2 were left in as a variable instead of an assumed numerical value.
The value of D n is therefore bounded and is a continuous function of co 2
for all finite and positive co 2 . A plot of D n versus co 2 must be of the form
shown in Fig. 73. The curve crosses over the abscissae at n places, including
the origin. These correspond to the natural frequencies. At the origin
the curve must have a positive slope, since for a very small but positive
o) 2 the sequence of computation must lead to a positive D n . Hence in our
17 We are assuming that generally we want to find the lowest natural frequency first.
18 This equation may differ from (16) only by a multiplication constant.
(3.13)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
215
ndicates that
trial to locate the lowest nonzero frequency a positive D
the trial value is too small and vice versa.
In practical applications Holter's method is most widely used for
analyzing the torsional vibration of the crankshaft of a multicylinder
engine connected to a generator or other driven systems. In such appli
cations there are usually two or three large inertia elements representing
generator rotor, flywheel, damper inertia, etc., and a number of small
Figure 73
inertia elements representing the pistons, connecting rods, and crankshaft
mass of the cylinders. To obtain an estimate of the lower natural frequen
cies of such a system, we may simplify it by "lumping" the smaller moments
of inertia with the larger ones near to them. Then the simplified system
has only one or two nonzero modes whose frequencies can be determined
quickly with methods described in the preceding chapters. These fre
quencies may now serve as the starting values for Holzer's method.
Holzer's method may also be used alternately with Rayleigh's method.
After a trial run Rayleigh's quotient for the amplitude ratio can be obtained
by the following simple relation:
o/2 = R(^ l9 </> 2 , • • • , </> n ) =
Hk A(/> 2 _
2/^ 2 "
M + D 2 E 2 +
• • + D^E^W
(^Q + B 2 C 2 + ■ ■ • + B„C„)
in which the symbols refer to items in Table 1. If D n = 0, co' naturally is
equal to co, and the correct frequency is obtained. If D n is small, co' is a
better approximation than co and can be used for the next run of computa
tions by Holzer's method. If D n is large, a corrective procedure, which is
described later, may be needed.
216
THEORY OF MECHANICAL VIBRATION
(3.13)
We shall now work an example. Let the crankshaft of a 4cylinder
engine and the attached inertia elements be equivalent to the system shown
in Fig. 74a. The values of the 7's and k's are
I x = 150
7 2 =20
h = h = h = h = 2
7 7 = 10
/c 12 =l
^23 = "
^34 = ^45 = ^56 = ^
^67 = 4
All are in their proper units, so that the unit of co will be 10 z radians per
second.
Ko
6
s 67
(a)
I/= 150
72 =24
/,' = 14
,'=1
K 23 — i7
Figure 74
If we lump 7 3 and 7 4 with 7 2 and 7 5 and 7 6 with 7 7 , we have the approxi
mately equivalent system shown in Fig. 74b, in which
// = 150 7 2 ' = 24
h' = 14
k' = 1
^23 — 1,1,1.1.1
613 + 3^3^4
12
17
By substituting these into (53) of Art. 2.9(a), we obtain the following
frequency equation :
12/' 14
F7 1+ 24
=
(3.13)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
217
This quadratic equation gives
co 2 = 0.0262 or 0.1000 (10 3 rad/sec) 2
We can now proceed to construct Holzer's table, Table 2, with co 2 =
0.0262.
Table 2
a> 2 = 0.0262
A
B
C
D
E
F
Ico 2
<f>
IC0 2 <f>
kAcf>
Acf>
w
3.930
1.000
3.930
3.930
3.930
1
0.524
2.930
1.536
2.394
0.399
0.1666
0.0524
3.329
0.175
2.219
0.740
0.3333
0.0524
4.069
0.213
2.016
0.672
0.3333
0.0524
4.741
0.248
1.768
0.589
0.3333
0.0524
0.2620
5.330
5.702
0.280
1.495
1.488
0.372
0.250
0.007 =
D 7
Since the value of D n in our first trial is sufficiently small, we have the
first natural frequency of the system; co^ 2 = 0.0262 x 10 6 .
Of course, we cannot always be lucky enough to have the estimated
answer so close to the true answer. For the purpose of illustration, let
us now work the same problem with a much poorer initial estimate.
Suppose by some scheme we obtain an initial estimate of co 2 = 0.0328.
This estimate results in Table 3.
Since D n = —1.084 < 0, we know that the trial co 2 is too large. We
can decrease this value and try again. A better way would be to compute
Rayleigh's quotient with the approximate modal vector represented by
column B of Table 3. According to our previous discussion,
D X E X + D 2 E 2 + ■ • • + D % E %
CO * = CD*
B 1 C 1 + B 2 C 2 + • • • + B 1 C 1
28.43
35.74
x 0.0328 = 0.0261
Thus we see that Rayleigh's quotient of the modal vector obtained from a
roughly estimated value of co 2 gives remarkably accurate results.
218
THEORY OF MECHANICAL VIBRATION
(3.13)
The modal vector obtained from column B of a first trial generally does
not satisfy the constraint relationship 2/c/> = 0, since D n ^ 0. It therefore
contains a rigidbody motion. If D n is not too large, we may disregard
this discrepancy. If D n is moderately large, a correction may be necessary.
Table
3
w 2 = 0.0328
A
B
C
D
E
F
/CO 2
<f>
Io> 2 <t>
k^
A0
\\k
4.920
1.000
4.920
4.920
4.920
1
0.656
3.920
2.572
2.348
0.392
0.1666
0.0656
4.311
0.282
2.066
0.689
0.3333
0.0656
5.000
0.328
1.738
0.579
0.3333
0.0656
5.579
0.366
1.372
0.457
0.3333
0.0656
0.328
6.036
6.280
0.396
2.060
0.976
0.244
0.2500
1.084 =
D 7
The correction consists of deducting from the denominator of Rayleigh's
quotient a term representing the kinetic energy of the zero mode. This is
permissible because of the orthogonality relation of the modal vectors.
The correction term is the first term of the denominator of (39). For the
present problem it is
€ = (A + h H T 4) <£mean
in which
, _ /A + / 2 < / > 2 + ••• + /„<£„
*n*an ~ ^ + ^ + . . . + j
In terms of quantities in the tabulated computation, we can see easily that
DJ
(A 1 + A t + ~' + A n )co<
Applying this formula to the example of Table 3,
1.084 2 _ 0.164
€ ~ 7.162co 2 ~ co 2
(3.13) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
Rayleigh's quotient corrected is then
28.43
219
CO
(35.74  0.16)
0.0262
although in this example this correction is too small to warrant the effort.
To obtain the second nonzero mode, we begin by assuming oj 2 = 0.1000.
The computations are tabulated as Tables 4 and 5.
Table 4
co 2 =0.1000
A
B
C
D
E
F
15.000
1.000
15.000
15.000
15.000
1.000
2.000
14.000
28.000
13.000
2.166;
0.1666
0.200
11.833
2.367
15.367
5.122
0.3333
0.200
6.711
1.342
16.709
5.570
0.3333
0.200
1.141
0.228
16.937
5.641
0.3333
0.200
4.505
0.901
16.036
4.009
0.2500
1.000
8.514
8.514
7.522 =
= D i
ZDE = 586
o>' 2 =
ZBC = 526 D 2 \
586
^—^(0.1000) =0.11
S/l = 3
3
By this time it appears very probable that the iteration process will
approach the answer from below. Rayleigh's quotient in Table 5 is most
likely to be very near but slightly lower than the true answer. The first
trial results in a 13 per cent increase and the second trial, a 5 per cent
increase. For the final trial we will use an "anticipated" answer, which is
about 2 per cent larger.
This time, D 1 is positive but very small, so that we know the true answer
is slightly below 0.121.
220
THEORY OF MECHANICAL VIBRATION
Table 5
(3.13)
CO 2 =
0.113
A
B
C
D
E
F
16.95
1.000
16.95
16.95
16.95
1.000
2.26
15.95
36.05
19.10
3.37
0.1666
0.226
12.58
2.84
21.94
7.31
0.1333
0.226
5.27
1.19
23.13
7.71
0.1333
0.226
2.44
0.55
22.58
7.53
0.1333
0.226
9.97
2.25
20.33
5.08
0.2500
1.130
15.05
17.05
3.28 =
D
S
a
DE = 964
964
/ 2 = X
914
25C = 914;
0.113 =0.119
Table 6
co 2 =0.119 x 1.02 =0.121
A
B
C
D
E
F
18.15
1.000
18.15
18.15
18.15
1.000
2.42
17.15
41.50
23.30
3.88
0.1666
0.242
13.27
3.21
26.51
8.84
0.3333
0.242
4.43
1.07
27.58
9.19
0.3333
0.242
4.76
1.15
26.43
8.81
0.3333
0.242
13.56
3.28
23.15
5.79
0.2500
1.210
19.35
23.40
+0.25 = D 1
(3.14)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
221
3.14 Electrical Analog of Mechanical Systems
and Electromechanical Systems
(a) electrical analog
We have emphasized on different occasions that vibratory systems may
have entirely different physical appearances, yet their motions are governed
by common mathematical equations, so that it is not always necessary to
analyze each individual system having a different makeup. We may go
one step further and observe that the differential equations of motion
C R L
— II — vwv\ — nmrr^.
(J eft)
Figure 75
which describe vibratory phenomena also govern the behavior of many
other types of systems. These systems are called analogous systems, and
they also come under the same treatment. The analogous systems most
closely related to mechanical vibratory systems are electrical networks.
Except for a different placing of emphasis, network theory is almost
indistinguishable from vibration theory. Historically, these theories have
always been borrowing results from each other.
Consider the simple circuit shown in Fig. 75. It consists of a resistance
R, an inductance L, a capacitance C, and a timedependent voltage source
e(t) connected in series. If i(t) denotes the loop current, Kirchhoff's law
will lead to the following integrodifferential equation:
Li + Ri +
C
/<
dt = e(t)
(>)
The indefinite integral above contains an integration constant to be
determined by the initial condition. By denning
q(t)
\
idt
we can write this integrodifferential equation as
Lq + Rq + q = e(t)
(ii)
(iii)
222
THEORY OF MECHANICAL VIBRATION
(3.14)
This equation is then the same as that for the forced vibration of a single
degreefreedom system.
rnq + cq + kq =f{t)
The electrical circuit in Fig. 75 is thus an analogous system of a spring
massdamper system in the sense that a timedependent voltage source will
produce a current in the circuit in an analogous way as a timedependent
force gives a velocity to the mass of a springmassdamper system. This
analogy is often called the direct analogy or the forcevoltage analogy.
(j)i(t)
C
Figure 76
A different electrical analog of the same mechanical system is the circuit
shown in Fig. 76. There we have a capacitance, a resistance, and an
inductance connected in parallel to a timedependent current source i(t).
Let v(t) be the voltage drop across the three parallel elements. The branch
currents flowing in these elements are, respectively,
v J If
i c = Cv i R = — and i L = — v dt
R LJ
Since the sum of the branch currents must be equal to the current from the
current source, we have
1 1 f
Cv +  v +  \v dt = i(t) (iv)
This integrodifferential equation has exactly the same mathematical form
as the one before, and because the indefinite integral contains an arbitrary
constant this integrodifferential equation can also be changed into a
differential equation of the second order by defining a function 19 q{t) as
/■
q(t) = \v(t)dl
cq + p + \i = m
19 This definition is intended to have mathematical meaning only. Although the
integral \i dt previously represented the physical quantity, electrical charge, this
physical meaning is not essential to our discussion.
(3.14) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 223
This analogy is called an indirect analogy or & forcecurrent analogy. Here
the voltage response to a current source is analogous to the velocity
response of a mechanical system to an applied force.
The foregoing simple examples show that when we speak of analogous
systems we must have in mind some specific phenomena with which the
analogy is to be drawn. In the present case the phenomena in question
are the signalresponse relationships between analogous pairs of physical
quantities.
We must realize also that both the mechanical system and the electrical
systems used for illustrations are merely models of other systems to which
they are analytically equivalent. Hence the analogy exists not only between
two specific systems but between two classes of equivalent systems.
An electrical analog to a mechanical system, which consists of a simple
arrangement of springs, mass elements, and dampers, can be conceived
by inspection without writing the differential equations of motion. There
are rules available whereby an analogous circuit can be drawn simply by
replacing mechanical elements with appropriate electrical elements and
mechanical arrangements with analogous electrical arrangements. These
rules permit a person who is more familiar with one type of system to
analyze the other type with which he is not as familiar. It is, however, not
our intention to analyze a mechanical system by converting it into a
network problem or to analyze an electrical circuit by vibration theory.
Our purpose of studying analogy is partly to create an awareness of its
existence and partly to enable us to build electrical analogs for complex
mechanical systems whose signalresponse relations can then be determined
experimentally on the analog. (It can be readily appreciated that experi
ments on electrical circuits are usually more economical than those on
mechanical models.) Therefore, in the following discussion it is assumed
that we know how to write the governing equations for both types
of systems when they are needed for the construction of analogous
systems.
Let us now study an example of a mechanical system having two
degrees of freedom, as shown in Fig. 11a. The differential equations of
motion are
WiA + c xl x Y + k u x ± + c 12 (x x  x 2 ) + k 12 (x 1 — x 2 ) = f x (t)
^22^2 ' ^22^2 • ^22^2 T ^12V^2 X l) ' ^12\ X 2 X l' =z j2\")
The two electrical analogs of this system are shown in Figs, lib and 77c.
The reader can verify that by applying Kirchhoff's law to the two loops
in Fig. lib the loop currents are governed by the same set of equations,
and by summing up the branch currents at the two nodes in Fig. 77c the
224
THEORY OF MECHANICAL VIBRATION
nodal voltage are also governed by the same equations,
tables help to describe the analogous relationships:
(3.14)
The following
Mechanical System
D'Alembert's principle
Degrees of freedom
Generalized velocity
Generalized force
Inertia element
Damping element
Elastic element
Coupling elements
Stationary frame
ForceVoltage
Analog
Kirchhoff's law
Loops
Loop currents
Voltage source
Inductance
Resistance
1 /capacitance
Elements common
to two loops
ForceCurrent
Analog
Continuity law
Nodes (not including
datum node)
Nodal voltage
Current source
Capacitance
Conductance
1 /inductance
Elements between the
nodes
Datum node
The electrical analogs of mechanical systems, which consist of masses,
springs, and dampers connected in a simple way, can be constructed
according to the analogous relations given in the foregoing tables. For
others, in which the couplings among the different degrees of freedom are
not obvious by inspection, it is necessary that the differential equations of
motion be obtained first. Take for instance the system shown in Fig. 53
and studied in Art. 2.11. There are no readymade rules by which we can
construct the electrical analogs for such a system without first writing
down the differential equations of motion; these equations were found
to be
(M + m)q\ + Kq 1 — m cos 0q 2 — Kq 2 =
in which
— m cos Bq x — Kq x + mq 2 + (K + k)q 2 =
q Y — X and q 2 = x
For the purpose of constructing electrical analogs we rearrange the
equations to read
(M + m — m cos 6)q\ + m cos d(q\ — q 2 ) + K(q 1 — q^ =
(m — m cos 0)q 2 + kq 2 + m cos 6(q 2 — q x ) + K(q 2 — q x ) =
From this set of equations we deduce that the two electrical analogs are
as shown in Fig. 78.
(3.14)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
225
"11
x/VVVV^—
\(t)
in
Voltage
source
W////m,
«12
•22
(a)
W///M//,
® cz
*2
(b)
^22 ^
v/VVV\A— 
fel1 dn m 22 ^2:
r^WHh^rK^Mh J vVrn
i x^ (uah
Voltage
source
l
4AAAH
M " 12 £w)
VA^Current Jh^
v =
(c)
Figure 77
(Datum node)
For a more complex example let us consider a mechanical system whose
motion is described by
Aq + Bq + Cq = f
x 10
From the signs of the offdiagonal elements of these matrices we observe
that in a forcevoltage analog for this system the loop currents must be
r 10 5 ii
[9 43"
[21 3
11]
A =
5 8 2
B =
4 12 6
C =
3 17
4
12 7
3 6 11
11 4
15
226
THEORY OF MECHANICAL VIBRATION
(3.14)
arranged to flow in such a way that there will be positive couplings between
loops 1 and 2 and loops 2 and 3 and negative couplings between loops 1
and 3. The arrangement is shown in Fig. 79. The values for the coupling
inductances, resistances, and capacitances are given by the absolute values
of the appropriate offdiagonal elements of the matrices A, B, and C.
(a)
(M + m  m cos 6) k (mm cos 6)
r^TRRRP
17
m cos 6
nffiw^i
12
(b)
v x (t)
Q
+
m cos 6
i—npm
1
K
v 2 (t)
— o
TS
I
,3
k
(O
Figure 78
Additional electrical elements are then placed in the loops to make the
total inductance, resistance, and capacitance in each loop equal to the
values of the appropriate diagonal elements of the matrices. Voltage
sources are placed in each loop to fit the column matrix f. The polarities
shown in Fig. 79 are for positive values of the elements in f.
The construction of a forcecurrent analog of the same system follows
the same idea. It is left to the reader as an exercise. (See Exercise 3.21.)
The types of analog circuits described so far contain only passive circuit
elements, which are capacitances, resistances, and selfinductances with
■
(3.14)
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
227
positive values. If these are the only kinds of circuit elements that may be
used, it is not always possible to construct analog circuits from a set of
differential equations obtained from a real mechanical system because to
do so may require circuit elements with negative values. On the other
hand, there are electronic devices that can be made to behave like negative
Cl3 #13 Lis
L12
L>23 #23 ^23
£33
x 12 = 5
^13= 1
L 23 = 2
X n = 1051=4
R n =943=2
C u =  10 /  = > > !(>
C 12 = \ X 10 6
C 13 = A x 10 6
,c, =  x 10 6
x 29
52=1
Roo = 12  6  4 = 2
10
21
3  11
Co
17
Figure 79
= 10
/?!<> = 4
/? 13 = 3
/? 23 = 6
•L 33 = 721=4
/?„ = 1 1 63=2
10 6
k 33
15_11_4
(short circuit)
capacitances, resistances, and inductances. These devices contain energy
sources in themselves and therefore are not passive elements. They are
commonly used in analog computers, which are nothing but electrical
analogs designed with a greater flexibility of operation. 20
(B) ELECTROMECHANICAL SYSTEMS
The analogy between vibratory mechanical systems and electrical
circuits is a particularly intimate one because in both types of systems the
phenomenon of interest is the transformation of energy forms viewed
20 For a brief account of analog computers see Engineering Systems Analysis by R. L.
Sutherland, AddisonWesley Publishing Co., Inc., Reading, Mass., 1958, Chapters 7
and 8.
228 THEORY OF MECHANICAL VIBRATION (3.14)
against the background of time. In a pure mechanical system the funda
mental relation is
(T + V) + P= W
at
where P is the timerate of dissipation of mechanical energy, 21 which is
usually in the form of heat, and W is the timerate of work done by
external forces. Similarly, in an electrical system we have the energy
relation
(S+U) + P=W (v)
at
where S = electric energy
U = magnetic energy
P = timerate of (heat) dissipation
W = input from external power source
Consider the simple circuit shown in Fig. 75 as an example. In terms of
the loop current i the energy expressions are
S =  q  U = Li 2 P=Ri 2 and W = ei
2 C 2
where q is defined by (ii). If we substitute these expressions into the energy
equation (v) and cancel out an i, the result is the equation from Kirchhoff's
law (i):
Li + Ri + \idt = e(t)
Now take the circuit shown in Fig. 76. In terms of the variable v, in this
case the energy expressions are
= 1 Cv 2 U = — I I v dt I P =  and W = iu
2 2L \J f R
and the energy relationship leads to the nodal current equation (iv):
1 1 r
cv + jv + \vdt= i(t)
21 The function P is the same as 2F, F being Rayleigh's function described in Art. 3.8.
(3.14) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 229
It is well known that mechanical energies and electromagnetic energies
from the point of view of thermodynamics are reversibly convertible.
Hence a mechanical system may be coupled to an electrical system to form
a socalled electromechanical system. The first law of thermodynamics
for such a composite system can be written
[ T + V+ S+U] + P=W (vi)
at
where P becomes the heat dissipation and W becomes the work input.
The term "work" is to be interpreted in the general thermodynamic sense,
which includes energy input by virtue of Newtonian forces as well as
electromotive forces. The generalized coordinates for an electro
mechanical system consist of geometrical variables (displacements, angles,
etc.) for the mechanical part and electrical variables (loop currents and
nodal voltages) for the electrical part. Where the two parts are coupled,
energy conversion takes place and imposes a relation between a pair of
variables, one mechanical and the other electrical. This relationship
serves as an equation of constraint, which describes the dependence of
one on the other.
An energy transfer between a mechanical phenomenon and an electro
magnetic phenomenon can take place in a number of ways. The most
important is that which takes place when a currentcarrying electrical
conductor moves in a magnetic field. Take, for example, the simple case
of the moving coil instrument studied in Art. 1.15. The basic construction
can be schematically represented by Fig. 80.
If we let = angular deflection of the meter coil
/ = current flowing in the coil
J = moment of inertia of the coil
k = constant of the restoring spring
R r/ = resistance of the coil
R = resistance external to the coil
e = externally impressed emf
the energy expressions are then
T = \Jd 2 V = \kd 2
P = i\R + R g ) and W = ei
230
THEORY OF MECHANICAL VIBRATION
(3.14j
We assume that the capacitance and the selfinductance of the coil are
negligible and so is the viscous damping. By substituting these into the
energy equation (vi), we have
(j6 + k0)0 + i\R + R g ) = ei
(vii)
This equation contains both the geometrical variable and the electrical
variable /. These, however, are related by an equation of constraint, which
Flexible leads
Winding
Figure 80
may be found by the following reasoning, based on Faraday's law. As
the coil turns, a back emf e b is produced, which is proportional to the
angular velocity 0.
e h = BO (viii)
in which B is determined by the field strength of the pole pieces 22 and the
geometry of the coil design. By applying KirchhofTs law to the loop,
the relationship between and i is then
e  e h = e  BO = i(R + R g ) (ix)
By eliminating the variable i between (vii) and (ix), we obtain finally
J6 +
B 2
6 + kO
B
R+ R g
R + R g
e(t)
(x)
22 It is generally the case that the magnetic field produced by the pole pieces is so
strong and the coil so designed that the field strength is undisturbed by the current
flowing in the coil.
(3.14) SYSTEMS WITH SEVERAL DEGREES OF FREEDOM 231
As it was pointed out before, the dependence relation between 6 and i
originates from the process of converting mechanical energy into electrical
energy, or vice versa. From this point of view we see that if the torque
produced in the coil by the interaction between the current and the
magnetic field is denoted by T b the rate of energy conversion is given by
Tfi = e b i (xi)
Hence the torque per unit current flowing in the coil is
— = "J = B ( X1
I U
The dynamic equilibrium equation of the coil is
JS + kd = T b = Bi (xiii)
Elimination of i between (xiii) and (ix) results again in (x). Incidentally,
(x) is the same equation as (i) of Art. 1.15 because of (xii).
Although a moving coil instrument is a very simple electromechanical
system, the analysis of its operation illustrates adequately the principles
available for studying the class of system in which the conversion of
mechanical energies into electrical energy, or vice versa, is effected by the
motion of a conductor in a magnetic field. There are other types of
conversion mechanisms which cannot be discussed here. Most of these
involve highly nonlinear conversion relationships.
Exercises
3.1. A bent wire is contained in a vertical plane. Its shape is described by
y = ax 2 for x >
(R  yf = /?2 _ r 2 for x <
A small ring slides on the wire without friction. What must the relation be
between a and R so that the system may be approximated by a linear system for
a small interval of x near x = 0?
3.2. Write the energy expressions for the systems shown, using as generalized
coordinates the quantities indicated. How are these expressions "linearized"
for small oscillations? (The correct word is perhaps "bilinearized, ,, but it is
awkward.)
232
THEORY OF MECHANICAL VIBRATION
(b)
Exercise 3.2
3.3. Rewrite the following matrix equations first in tensor notation, then in
longhand for n = 3.
A f + ag = h g A B f = /.
3.4. Rewrite the following equations in matrix notation:
3.5. Write out in longhand the determinant in (16).
3.6. Illustrate equations (12) to (16) with a numerical example using
■2"
C =
A =
21
7 11
7
19 7
1
7 5_
14 3
3 6
2 1
1
2 J
x 10 3
Solve the frequency equation obtained and determine the modal vector corre
sponding to each frequency.
3.7. To which of the equations in Chapter 2 are the following equations in
this chapter equivalent: (12), (13), (14), (15), (16), (17), (19), (20) and (21)?
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
233
3.8. By matrix iteration, determine the smallest eigenvalue and the associated
eigenvector of the problem in which
A =
5.83
1.21 3.12 
ri3 3
7
1.21
4.60 2.57
C =
3 9
4
3.12
2.57 6.95
7 4
5
x 10 4
3.9.
(a) Find a second eigenvector for Exercise 3.8 by the method described in
Art. 3.11(d).
(b) Reduce the order of the problem of Exercise 3.8 to n = 2 by the method
described in Art. 3.11(e).
3.10. Show that a set of mutually orthogonal vectors must be linearly inde
pendent. That is
o^r* 1 ' + a 2 r (2) , • • • , oc n r in) ^
unless a x = a 2 , . . .
3.11. Prove that
<2(u) > ^
by the fact that both the denominator and the numerator in (36) or (38) must be
positive for arbitrary u.
Hint. Show first that the elements of the diagonal matrix M are always non
negative.
3.12. Derive equations (47) and (48) in Art. 2.8.
3.13. Set up the differential equations of motion for the beam shown.
■EI
^
a3
Exercise 3.13
3.14. Find the steadystate solution of
A q + C q = ie ia} f l
in which A and C are given in Exercise 3.6, and
f  [3, 4, 5] o>, = 20
(a) by method discussed in Art. 3.7(b), (b) by methods discussed in Art. 3.7(c).
234
THEORY OF MECHANICAL VIBRATION
3.15. Find C for the system shown. Does C _1 exist for this system? Can
any of the influence coefficients be defined ?
M,I
3.16. Given a symmetrical matrix A and a set of linearly independent vectors
(a) Show that the set of vectors v (1) , v <2) , . . . , v (n) defined below are mutually
orthogonal with respect to A.
>(1> •.<!)
(2) _ r <2) _
(3) _ <3)
v (1 >A r (2)
v ,2) A r (3 '
v °' = r
An) _ „(n)
v (1) A r (3;
v (2) A v (2) v (1) Av (1)
(i)
y (nl)^ r (n)
;(nl)j^ v (nl)
r(nD
v (1) A r (n)
v (1) Av (1)
,(1)
(b) Show that if the r's satisfy
Cr=AAr A^O
the v's defined are also mutually orthogonal with respect to C.
3.17. Verify the statement following (52).
SYSTEMS WITH SEVERAL DEGREES OF FREEDOM
235
3.18. Make up an eigenvalue problem for which the answer is
rr
r i I
r i ~
r i
2
2
r (2) _
i
2
r (3) _
2
2
I«(4) _
2
1
_i_
_2^
1 .
_l.
X l = 1 A 2 = 4 A 3 = 8 A 4 = 20
Note: The answer will not be unique.
3.19. Find the lowest two natural frequencies by Holzer's method for the
following torsional system
h
h
h
h
h
h
unit
200
10
10
10
10
40
lbftsec 2
^12
^23
^34
^45
^56
unit
1000
8000
8000
8000
4000
lbft
3.20. Draw the forcevoltage and the forcecurrent analogs of the mechanical
systems shown.
(a)
M,I,R^
AMAAr
k
CHAPTER 4
Vibration of Elastic Bodies
4.0 Introduction
Vibratory systems having a finite number of degrees of freedom are
only idealized models of certain real systems. The essential property
of such systems is that the inertia forces are those due to a finite number
of rigid bodies, and the potential and dissipative forces are the interactions
between pairs of these rigid bodies. We have, therefore, rigid bodies,
which can store 1 only kinetic energy, and weightless bodies and force
fields, which can store only potential energy. Real systems consist of real
bodies that are neither perfectly rigid nor completely without inertia,
but they may be considered approximately one way or the other under
certain circumstances.
The validity of such simplifying approximations depends upon the
relative rigidity and amount of mass in the different parts of the system
as well as the nature of the vibratory motions being considered. For
instance, in order for a simple springmass system to be considered a
singledegreefreedom system, the mass must be relatively rigid and heavy
in comparison with the spring, and the motion must be such that the
amplitude of the mass is of the same order of magnitude as that of any
portion of the spring. The last stipulation is necessary because the system
may be set into high frequency vibrations by "stroking" the spring. In
such vibrations the mass remains nearly stationary, while the different
parts of the spring vibrate, and the system can no longer be considered as
having only one degree of freedom.
1 The concept of energy stored in bodies is sometimes convenient but cannot bear
close scrutiny. The meaning of the word "stored" is not to be taken too literally in the
present context.
236
(4.1) VIBRATION OF ELASTIC BODIES 237
In this chapter we discuss the vibration of elastic bodies, which cannot
be considered as rigid or as without inertia. The configuration of the
systems containing such bodies thus cannot be described by a finite number
of generalized coordinates. The analysis of these systems is naturally
more complex. The increased complexities, however, lie mostly with the
mechanics of elastic deformation (theory of elasticity) and the mathematics
of solving boundaryvalue problems. There are no general methods in
dealing with these two aspects of the problem. On the other hand, the
conceptual matters related to theory of vibration can be discussed by a
general treatment. This treatment we shall emphasize. Complete solutions
for a few simple problems are carried out as illustrations. With these
examples it is hoped that the reader will have enough understanding to
analyze more involved problems, once he has also acquired the necessary
knowledge of the theory of elasticity and the necessary mathematical
tools.
4.1 Coordinates and Constraints
In order to describe the configuration of an elastic body, it is necessary
to specify the spatial location of every mass particle in the body. The
most direct method of description is then to use the displacement of the
particles as the coordinates. But the socalled mass particles in an elastic
body are not discrete. They have a continuous distribution in space. 2
Therefore, it is not possible to label them by integer indices or to assign
finite values for their masses.
Let x, y, and z be the Cartesian coordinates of a generic point P in an
elastic body in static equilibrium. Let p(x, y, z) be the mass density at P.
A generic mass element dm = p(x, y, z) dV then plays the role of a mass
particle. Let w, v, and w be the three components of the displacement of
this mass element from its equilibrium location. These components are
both space and time dependent.
u = u(x, «/, z, t)
v = v(x, y, z, t)
w = w(x, y, z, t)
These displacement functions 3 can be considered as the coordinates of the
system. The space variables x, y, and z are used to designate the particular
2 We are looking at things macroscopically, not microscopically.
3 Those who have studied fluid dynamics should note that they furnish essentially
the Lagrangian description of the motion of a deformable body.
238 THEORY OF MECHANICAL VIBRATION (4.1)
element being considered. These variables play the role that was played
by subscripts before.
Since there are infinitely many combinations of x, y, and z, each corre
sponding to a mass element, there are infinitely many degrees of freedom in
the vibration of an elastic body. At first glance it appears that the number
of degrees of freedom is not only infinite but noncountable, since the mass
elements in a continuous medium are not countable. 4 This, we shall see,
is not the case unless the body is infinite in extent. The question of
countability is of interest because we have seen that with each degree of
freedom there is associated a natural frequency. If the degrees of freedom
are countable, the natural frequencies will be discrete; that is, we can list
them in an infinite sequence m v co 2 , . . . , co„, ....
The coordinates u, v, and w, however, are not completely independent.
They are subject to certain constraints. At the boundary where the body
comes into contact with its support or with another medium the displace
ment functions are often subjected to certain restrictions, which are called
boundary conditions. In the interior of the body other types of restrictions
prevail, such as the requirement that the displacement function be con
tinuous and differentiable a certain number of times with respect to space
variables. Restrictions also arise out of assumptions. For instance, in
specifying the deformed shape of a slender beam, we may assume that the
points on a given cross section move together as a rigid plane.
Because of the different kinds of constraints described it is often possible
to express a displacement function as the sum of an infinite series of given
functions. For instance, in describing the shape of the elastic curve of a
deflected beam simply supported at the ends, we may use the trigonometric
series
™ nnx
w = 2 q„ sin —
n = \ L
By specifying a set of values for the coefficients q lf q 2 , ■ . . , q n which
are infinite but countable, the configuration of the beam is uniquely
specified. Hence the set of </'s may be considered as the generalized
coordinates of the system. This method of describing the configurations
of elastic bodies is important because it allows us to borrow the results
from the analysis in Chapter 3.
4 By "countable" or "denumerable" we mean that the set of things can be put into
onetoone. correspondence with integers 0, 1, 2, 3, ... . It can be shown, for instance,
that rational numbers are countable, whereas the number of points on a line segment is
not.
(4.2) VIBRATION OF ELASTIC BODIES 239
4.2 Formulation of a Problem by Differential Equation
The direct application of Newton's second law to a mass element
results in
P^dV=df x
P^ dV = d fy
P 1 f 2 dV=df z (1)
in which df x , df y , and df z are elemental forces acting on the elemental
volume dV. In free vibrations these are elastic forces caused by local
deformations of the body. They are related to u, v, and w by the laws of
elastic deformation, which relate forces to stresses, stresses to strains, and
strains to displacements. General equations are available from theory of
elasticity to express these forces in terms of displacements. 5 Since such
general equations are seldom solved in practice, we shall dispense with the
exact expressions here. It is sufficient to say that the equations are of the
form
B 2 u
d?
P*72 = L *( w ' y > w )
P^79 = L <>> v > w ) ( 2 )
in which the L's are linear differential operators involving partial deriva
tives with respect to space variables of different orders.
■ / d d d i \
L = L — , — , — , elastic constants
\ox ay dz i
It is to be noted that time t does not enter explicitly at the righthand
side.
The partial differential equations so obtained are merely statements of
Newton's law of motion and Hooke's law of elastic deformation, with
certain references to the mass distribution and elastic property of the
material that makes up the elastic body. These equations are, however,
5 These equations are associated with the name of Navier.
240 THEORY OF MECHANICAL VIBRATION (4.2)
not specific enough to define a vibration problem. They must be supple
mented by other conditions. The first set of conditions is called the
boundary conditions, which describe how the elastic body is supported and
how it comes into contact with other media along its boundary. Differen
tial equations of motion, together with a set of boundary conditions,
constitute a boundaryvalue problem. Such a problem specifies the types of
vibratory motions a given elastic system may have. It is often the problem
of practical interest. If, however, the problem is to find out how a given
body actually vibrates, we shall also have to know how it was set into
vibration to begin with. This information is furnished by socalled
initial conditions, which specify the displacement and velocity distributions
at a certain instant / = t .
Let us now illustrate the discussion with a few examples.
(a) longitudinal vibration of a slender bar
and mathematically analogous vibration problems
During longitudinal vibration each elemental length of the bar undergoes
compression and extension alternately. If the bar is relatively slender, the
inertia forces in the transverse directions, due to the contraction and
expansion of the cross sections, are negligible. The internal forces in the
bar are then essentially axial. The stressdisplacement relationship in a
bar subjected only to axial forces is given by Hooke's law as
(y x = Ee x = E— (3)
ox
in which the xaxis is parallel to the axis of the bar, and a, e, and E are
the stress, strain, and Young's modulus, respectively (see Fig. 81).
Consider now the equilibrium condition for an elemental length of the
bar between two cross sections spaced dx apart. Application of Newton's
second law results in 09 ~ , ~ N
o l u o I ou\
<> a w* = tA ea tJ (4)
For bars with uniform cross sections the equation simplifies into
d 2 u_Ed 2 u
dt 2 ~~ P dx~ 2
Since there is only one space variable, namely x, the mathematical
model of the bar has only two boundary points corresponding to the two
ends of the bar. 6 Various boundary conditions are possible.
6 The boundary of the physical body itself naturally is a surface having an infinite
number of points. The mathematical model reduces the bar into a onedimensional
object.
(4.2) VIBRATION OF ELASTIC BODIES 241
At a fixed end the displacement is zero.
u =
At a free end the stress is zero.
dx
At an end at which a mass M is attached the inertia force of M and the
elastic force at the end of the bar must be in equilibrium.
d 2 « du
tit 1 ox
— ^ u(x, t)
^
1
°x A n ff xA+fe (<J x A)dx
dx
Figure 81
At an end at which a spring of constant K is attached, the two elastic
forces must be in equilibrium.
du
Ku + EA— =
ox
Each of these boundary conditions, when valid, is supposed to hold
only for one particular value of x but for all values of t.
All of these boundary conditions are linear homogeneous because they
contain only first power terms of the dependent variable wand its derivatives.
The linear differential equation (4), and a set of linear boundary conditions
together make the problem linear, and the superposition theorem holds;
it is easy to see that if u x (x, t) and u 2 (x, t) are two solutions to the boundary
value problem, their linear combination c x u x + c 2 u 2 is also a solution.
The boundary condition of the first type u = is described by the term
"imposed" or "geometrical ." The other types are called natural or dynamical
boundary conditions. The significance of this differentiation is discussed
in Art. 4.7(b).
242
THEORY OF MECHANICAL VIBRATION
(4.2)
The initial conditions needed to specify a particular solution of the
vibration problem are of the form
and
u(x,0)=f(x)
ffiL*
These conditions are linear but not homogeneous, since the righthand
sides are not zero. The statement of the superposition theorem as applied
to an initialvalue problem appears in Art. 1.10.
Uq
Uj
u 2
/\/\/\/v^/\/\/\/v4k/\/\/\/v^ #vW\A#y\/\/\/
"n + l
m x
m 2
m 3
m n _i
Figure 82
It is a worthwhile digression to compare the formulation of this problem
with that of a similar problem in which the mass distribution is not
continuous. Consider a system made up of n masses, m l5 m 2 m n ,
connected by n + 1 springs, k , k l9 k 2 , . . . , k n , into a continuous chain.
(Fig. 82.) If all the masses are constrained to move along a straight
line, with u lt u 2 , . . . , u n as their respective displacements, the equation of
motion for the rth mass is
d 2 u
Mi ^ = ki(u i+1  u t )  k^iu,  u^)
1,2, ••,/!
In writing this general expression, u and u n ^ x are taken to be the displace
ments of the outer ends of the two end springs. To use the notation of
difference equation, we set
Aw 2 = u i+1  u t
Am,_i = u . _ Uil
Mk, &u t ) = k t Au r  ki_ x Aw,_ x
Hence the general expression becomes
m/^ = Wt Au t ) i=l,2,,n
(4.2)
If all the k's are equal, we have
d 2 u, k
VIBRATION OF ELASTIC BODIES
243
dt 2 m,
A 2 w ? .
/= 1,2,
These difference equations are "images" of the partial differential equations
(4) and (5).
The description of this system, as given by the difference equations, is
not complete without some reference to the two end springs which terminate
the chain. For instance, if one end of k n is fastened to a fixed point, we
must also specify that
u n <* =
and, if one end of k n is completely free,
u m ^ — u„ = Aw,
These conditions correspond to the boundary conditions of the partial
differential equation.
Two other vibration phenomena have the same governing differential
equation as the longitudinal vibration of a bar. The first is the torsional
vibration of a shaft with a circular cross section. Let the axis of the shaft
be in the ^direction and the rotation of a generic shaft cross section be
denoted by
6 = 6(x, t)
The torsion formula states
30 __ M
dx JG
244
THEORY OF MECHANICAL VIBRATION
(4.2)
in which M is the torque, G the modulus of rigidity, and J the polar
moment of inertia of the cross section. On the other hand, Newton's law
of motion leads to
d 2 6 _ dM
P J ^2
Upon elimination of M, the resulting equation is
d 2 6
P J
dt 2
dx \ dx]
— H dx
M
(I
.)
M+dMdx
dx
V+^dx
dx
dt 2
Figure 84
For shafts with uniform cross sections
d 2 e
di 2
Gd 2 d
p dx 2
(6)
Except for interpretation of the symbols, these equations are identical
to (4) and (5).
The second analogous phenomenon is the transverse vibration of a
thin stretched string. Assume that the vibration is constrained within
the #2/plane. The configuration of the string during vibration is described
by (Fig. 83):
v = v{x, t)
(4.2) VIBRATION OF ELASTIC BODIES 245
with v = being the equilibrium configuration. The forceacceleration
relationship of an elemental length of the string in the ^/direction is
d 2 v
P A fai dx = dS v
in which S y is the ^component of the tension S existing in the string.
Neglecting secondorder terms of dv/dx, we have
dv
S y = Ssm6= S —
(4:)
dS y = ^\S :;,
Hence
d 2 v d i dv"
0*v o 0v\
pA o7* = dA s Tx)
This is a nonlinear relation, as S depends also on v. However, if the initial
tension is strong and vibration v small, S may be taken as a constant. We
then have
d 2 v a d 2 v
dt 2 p dx 2
where o = SjA.
(b) lateral vibration of a slender bar
When a slender bar enters into a transverse vibration, the stress and
strain distributions produced in the bar are approximately the kind
described by the elementary theory of beam for static bending under
lateral loads. That is to say, the inertia forces due to such things as shear
strain and anticlastic deformation of the cross sections can be ignored.
Let us choose a righthanded coordinate system such that the zaxis is
the centroidal axis of the bar and the zzplane is the neutral plane of the
bar during symmetrical bending. The deformed shape of the bar is then
described by the equation of the socalled elastic curve, which lies in the
#2/plane (Fig. 84) :
v = v(x, t)
in which v is the displacement of points on the centroid axis from its
unstressed position. For small deformation, the flexural formula states
d 2 v
Ox 1
246
THEORY OF MECHANICAL VIBRATION
(4.2)
in which M is the internal bending moment and /is the moment of inertia
of the cross section. 7 The elementary theory of bending states further
that in the absence of externally applied moment
dM
dx
=— = V = internal shear
and
dv
= fv
(8)
(9)
in which/^ is the distribution of externally applied force in the ^direction.
Figure 85
In a dynamics problem f v is replaced by inertia force. Hence
d 2 v dv a 2 / d 2 v
P A
For bars with uniform cross sections we have
EI dh
pA dx*
(10)
(11)
Let us insert a few remarks about our assumption of slenderness of the
bar. In the deviation of (10) and (11) this assumption enters into a number
of places. As a rough guide we may say that most of the results from
elementary theory of bending are valid when the square of the depthto
length ratio of a beam, (c/L) 2 , is much less than one. This is so in static
bending, where the applied loads are predominantly in one direction and
7 The sign conventions used here for bending moments and shear are those commonly
adopted by modern textbooks on strength of materials. It is also permissible to consider
the sign conventions as being defined by (7) and (8).
(4.2) VIBRATION OF ELASTIC BODIES 247
the length of the beam enters as the distance between points of zero
deflections or of zero moment. In vibratory motions, however, a beam
may have deflections of the form /" "v or of the form AAAJ . Thus a
given bar may be considered as slender in one kind of vibration but not
in the other kind. The quantity L is therefore more appropriately inter
preted as the "wavelength" of the deflected form during vibration.
If the ratio c/L is small but not altogether negligible, whereas (cjL) 2
is negligible, the equation of motion may be corrected for the inertia
effect of the rotation of the cross sections. The correction begins with (8).
dM
— — dx = V dx — moment of inertia force on dx
ox
or
dM , , a 2 (dv\ J
^dx=Vdx + P Iy)dx
Together with (7) and (9), this relation leads to
a 2 / d 2 v\ dH d*v
It is interesting to note that this equation is not in the general form of (2).
The reason for this deviation is that in our scheme of describing the
deformation of the bar we think of it as being made of elemental rigid
slices, each having a translation v and a rotation dv/dx. Hence the dis
placement function v is not really the displacement of a generic mass
element in the elastic body.
Depending on end conditions, appropriate boundary conditions must be
prescribed. A few of the frequently encountered boundary conditions
are as follows (Fig. 86):
At a fixed or builtin end,
v =
and
ox
a pinned end, where M 
17 =
= 0,
and
d 2 V
dx~ 2 = °
At a free end, where M = and V = 0,
/ d 2 v\
— 0 and 
dx 2 dx
Still other types of boundary conditions are possible. These are discussed
in specific problems.
248
THEORY OF MECHANICAL VIBRATION
(4.2)
For this problem those boundary conditions involving only v and dvjdx,
which describe the displacement and the slope at an end, are called imposed
or geometrical; those involving d 2 v/dx 2 or d 3 v/dx?, which describe the
moment on shear at an end, are called natural or dynamic.
The initial conditions required to specify a particular solution of the
vibratory motion are given by
*>Cm ) =f( x )
dv
dt
(?, t ) = g( x )
dx
3*2 ~ U
<F&
dx*)
0=°
Figure 86
:I
( £ '0) = °
(c) VIBRATIONS OF STRETCHED ELASTIC MEMBRANES
AND OF FLAT THIN PLATES
These problems are twodimensional extensions of the problems for
vibrating strings and beams. We shall not derive the differential equations
of motion here, since what is involved in the derivations belongs to the
theory of elasticity. It is not difficult, however, for readers to accept the
following results.
For transverse vibration of a membrane lying in ^plane when at rest
the differential equation of motion is
d 2 w /a 2 d*\
(13)
in which h is the thickness of the membrane and S is the surface tension in
the membrane, which is assumed to be constant. For a membrane
initially stretched to a constant surface tension the only boundary con
dition physically realizable is that
w =
on all boundary points. This equation, as well as its onedimensional
and threedimensional counterparts, is called the wave equation. These
(4.3) VIBRATION OF ELASTIC BODIES 249
equations are studied extensively in acoustics and in hydrodynamics.
In other applications the boundary conditions may also contain normal
derivative dw/dn, which is the rate of change of w along a direction normal
to the boundary of the system.
The equation for the vibration of a thin plate is the twodimensional
extension of that for a beam. It reads
d 2 w __ / a 2 ( jp
ph ~d? " w + df
V 2 (E'I'V 2 w)
i v h d 2 w\
(14)
in which E' — Ej(\ — v 2 ), v = Poisson's ratio, / = /z 3 /l 2, and h = plate
thickness.
4.3 Separation of Time Variable from Space Variables —
Reduction to Eigenvalue Problems
Most of the boundaryvalue and initialvalue problems used for illustra
tions in Art. 4.2 can be solved under a number of circumstances.
Before we get into the details of solving particular problems and thereby
lose sight of the overall picture, let us study our problem in general terms
for a little while longer. Without losing too much generality, we do wish,
however, to restrict the discussion to equations having only a single
dependent variable, 8 which we shall call q in all cases.
All of the equations used for illustrations in Art. 4.2 can be put into
the general form of
Mq+Lq = (15)
in which M and L are two linear differential operators involving only
space variables x, y, and z, and the dots over the q represent partial
differentiation with respect to time. For instance, in (12)
M  ( pA  pi S)
d 2
d 2 I d 2 \
and in (14)
M = ph
L = V 2 (£7'V 2 )
8 It is also possible to write a single vectorial equation to replace the three scalar
equations in (2).
250 THEORY OF MECHANICAL VIBRATION (4.3)
In form, at least, if in nothing else, (15) resembles (13) of Chapter 3.
This resemblance gives us ideas about possible subsequent developments.
As we recall, what we used to do in studying free vibrations was first to see
if the system could vibrate in such a way that all the coordinates varied as
a harmonic function of time. To this end we let
q.(t) = r, cos (cot  a)
Similarly, for the present problem, we may let
q(z, y, z, t) = r(x, y, z) cos (ojt  a) (16)
Introducing this assumption 9 into (15), we have
co 2 Mr + Lr = (17)
This is a differential equation containing no t, just as the matrix equation
(16) in Chapter 3 does not contain t. A more formal way of arriving at (17)
is by a method called the separation of variables. Here we let
(18)
and (15) becomes
or
Since the first ratio cannot contain t and the second cannot contain
x, y, z, neither ratio can contain any variable and / must be a constant.
Furthermore, it is not difficult to see that ?. must be positive so that p
will not be an exponential function that grows with time. Thus by setting
X = co 2 , we have
p + co 2 p =
Lr = co 2 Mr (19)
These two equations share in common only the parameter co 2 , whereas
both the dependent variables and the independent variables in them are
different. For every solution we can find for this pair we have a solution
for the original equation (15). But the reverse is not true, since a solution
of (15) may or may not be expressed in the form of (16) or (18). We can,
however, entertain hopes that since co 2 is yet undetermined there is more
than one set of solutions for (19) and the totality of linear combinations of
these solutions may include all the solutions of (15). This is again a
9 For the time being the possibility of a solution in the form ( 1 6) is only an assumption
yet to be proven.
q =
r&, y,
*M0
p\Ai
' + pLr
=
Lr
m7
= _P
P
= /
(4.3) VIBRATION OF ELASTIC BODIES 251
situation similar to that encountered in Chapter 3, where a particular
solution in free vibration is representable by a linear combination of
principal modes.
So far the discussion has been centered around the differential equation
of motion which does not by itself define a physical problem. The accessory
conditions concerning the initial values and the boundary values must also
be considered. Let us take up the boundary conditions first, since these
are an integral part of the problem that defines the possible ways in which a
system may vibrate. The boundary conditions can be expressed in the
general form
1%) =
in which N is a linear differential operator of the space variables and
sometimes also of the time variable /. If N does not contain /, the replace
ment of q by rp according to (18) will result in
N(r) =
because N(q) is linear and homogeneous in q. If N contains t, it is
because the boundary condition says something about an inertia force.
In this case / enters only in the form of d 2 /dt 2 . Since
if q is in the form of pr, the boundary M(q) leads to N'(r) where N' is
obtained by replacing d 2 /dt 2 in N with co 2 .
To recapitulate, we now have the problem of finding a function r(x, y, z)
that satisfies a linear homogeneous differential equation
L(r) = co 2 M(r)
and a set of linear homogeneous boundary conditions of the form
N(r) =
where co 2 is a parameter yet unspecified and N may or may not contain co 2 .
In many vibration problems in which M is merely a multiplication
factor that can be incorporated in L, the differential equation simplifies
into
L(r) = co 2 r
In any event, the problem is markedly similar to the eigenvalue problem
we dealt with in Chapter 3. We expect, and we shall find, that the situation
is as follows:
(i) The problem has a nontrivial solution only when co 2 takes on one of
a set of values called eigenvalues co 2 , to 2 , .... To each eigenvalue is
252 THEORY OF MECHANICAL VIBRATION (4.3)
associated a solution r t (x, y, z) called the eigenfunction belonging to a>?;
together they describe the frequency and the deformation of a natural
mode of vibration of an elastic body.
(ii) Since the degrees of freedom of an elastic body are infinite, there
are infinitely many natural modes. All motions of a given elastic body
in free vibration can be represented by a linear combination of its natural
modes.
Let us now analyze a few of the simpler problems illustrated in Art.
4.2.
(a) longitudinal vibration of a slender bar of
uniform cross sections with a concentrated
mass m attached to one end while being held
AT THE OTHER END (FIG. 87).
The differential equation and its boundary conditions are
E d 2 u _ d 2 u _
p dx 2 dt 2
At x = 0,
u =
At x = L,
d 2 u EA du „ rm ^
dt 2 M dx
Let /u be the ratio of the attached mass to the mass of the bar. Then
EA E
M = apAL — =
ry M fiLp
The eigenvalue problem results from letting u(x, t) = p(t)r(x):
d 2 r co 2 p n ,_.
* 2 + ir = (2I)
and
For simplicity let
r(0) =
2
r'(L)/*L^r(L) = (22)
P 2 = ^
h E
The solution of (21) is evidently
r = D cos fix + B sin fix (23)
(4.3)
VIBRATION OF ELASTIC BODIES
253
To satisfy r(0) = 0, D must vanish and p (or oj) must be real. To satisfy
the remaining boundary condition, we must have
or
B(P cos PL  juLp 2 sin &L) =
pPL = cot PL
(24)
This equation in p (or w) is the frequency or characteristic equation of the
system. It is equivalent to the determinant equation in the matrix eigen
value problem. The coefficient B in (23) is indeterminate. Without losing
generality, it may be taken as unity.
E,A,p
u(x, t)
M = [ipAL
Figure 87
The equation (24), being transcendental in nature, has an infinite number
of roots. The first few positive roots 10 for four values of /u are shown in the
following tabulation :
h L
hL
LI =
ir/2
3tt/2
5t7/2
7t7/2
977/2
11 77/2
t* = i
1.0769
3.6435
6.5789
9.6295
12.7223
15.8198
v = 1
0.9604
3.4348
6.4373
9.4258
12.6453
15.7712
Jil = 00
77
277
377
4tt
5tt
The condition /u = corresponds to a free end at x = L. In this case
Pi = (2/ — X)n\2L. The condition ju — 00 corresponds to a fixed end at
x = L; there ft = (/ — 1)tt/L. With the exception of these two extreme
cases, the eigenvalues are not spaced regularly apart. But, for all ju > 0,
Pi approaches (1 — X)tt\L rather quickly. This is understandable, since
the inertia force goes up with the square of frequency and at high frequen
cies the end mass becomes nearly immobile.
The negative roots represent nothing new when put into (23).
254 THEORY OF MECHANICAL VIBRATION (4.3)
Returning to the function p, we see that
p + j 2 p =
P
Hence
p = C cos (co? — a)
= C cos [V(EI P )Pt  a]
A typical solution to the boundaryvalue problem (20) is then
u(x, t) = r(»/?(/) = C sin /3a: cos [V(Elp)pt  a]
in which /? is any one of the solutions to the frequency equation (24).
Since the problem is linear and homogeneous, by the application of the
superposition theorem we obtain the following series solution to the
boundary value problem :
u(x, = 2 Q sin ft* cos [V(£/p)/V  «J (25)
i=\
The constants C and a are arbitrary as long as they produce a convergent
series, and the function u(x, t), so defined, is at least twice differentiable
with respect to both x and t in the ranges 11 < x < L and t > 0. With
the freedom of choice in the C's and a's, we hope to be able to match the
initial conditions of a given problem and to arrive at a particular solution.
Let the initial conditions be given by
u(x, 0) = / (x) and I — = g(x)
If we know how to expand these two functions into two infinite series of
eigenfunctions
u(x, 0) = f(x) =  /, sin fa
(26)
\ot/ t=Q ? =i
we can determine the constants in (25) by the coefficients f t and g { . By
putting t = in (25) and its derivative and comparing the results with
(26), we have
C cos a f = fi
,— (27)
V(E/p) ftC, sin a, = g<
11 This differentiability requirement is a little more stringent than mathematically
necessary.
(4.3) VIBRATION OF ELASTIC BODIES 255
For the special cases of ju = and /u = oo the eigenvalues are spaced
regularly apart; the two series in (26) are Fourier sine series and they can be
obtained in the standard way. For other values of /u the series in (26) are
no longer Fourier series in the ordinary sense of the term. The method of
expanding given functions into infinite series of eigenfunctions is discussed
in Art. 4.4.
(b) lateral vibrations of bars having uniform
cross sections
The differential equation of motion as derived in the last article is
*!£?* (28)
To separate the time variable from the space variable, we look again for
solutions of the type
v(x, t) = p(t)r(x)
The substitution of pr for v results in two ordinary equations:
p + co 2 /? = (29)
£/»0 (30)
in which
OA P A0)2
(31)
EI
The solution for (30) is
r = A sin fix + B cos fix + C sinh fix + D cosh fix (32)
The eigenvalue problem is now as follows:
Given a set of end conditions, determine the set of eigenvalues 12 of
P so that the constants A, B, C, and D will not vanish together, and find
the ratios A \B :C :D corresponding to the eigenvalues.
Consider a cantilever beam which is fixed at one end and free at the
other. The boundary values are (Fig. 88)
r(0) = r'(0) = r"{L) = and r m {L) =
in which L is the length of the beam. From the first two of these conditions
we have
B + D = and A + C = (33)
12 It is immaterial whether we call /S or co the eigenvalue, since they are definitely
related by (31).
256 THEORY OF MECHANICAL VIBRATION (4.3)
From the last two conditions we have
—A sin pL  B cos fiL + C sinh PL + D cosh 0L =
A cos pL + 5 sin 01 + C cosh 0L + D sinh 0L =
or
C(sin PL + sinh PL) + £>(cos 0L + cosh PL) =
C(cos pL + cosh 0L) + D(sinh pL  sin 01) =
1 jj^W
(34)
Figure 88
For nontrivial solutions the determinant formed by the coefficients of C
and D must vanish. Hence
sin PL + sinh PL cos PL + cosh PL
cos PL + cosh /5L sinh PL — sin /iL
or
cos PL cosh PL = 1 (35)
This is then the characteristic or frequency equation of the problem. It
has an infinite number of solutions. The first six are
P X L P 2 L P Z L p 4 L P 5 L p 6 L
1.875 4.694 7.855 10.996 14.137 17.279
To each value of P there is an co given by (31).
The eigenfunctions are obtained by substituting (33) and (34) into (32),
r(x) — sinh Px — sin px + y(cosh Px — cos fix) (36)
where
D sinh pL + sin PL
7
C cosh PL + cos pL
We thus have a set of eigenvalues and eigenfunctions representing the
frequencies and the shapes of the deformed bar of the natural modes.
Mathematically, because the boundaryvalue problem is linear and
homogeneous, its general solution can be represented by an infinite series
of the eigenfunctions that is convergent in the interval < x < L.
00
v(%, = 2 Q cos ( C( V — <*t) r i( x ) (37)
(4.3) VIBRATION OF ELASTIC BODIES 257
To find the particular solution under a given set of initial conditions,
v(x, 0) = f(x) and v(x, 0) = g(x)
we first expand f(x) and g(x) into two infinite series of eigenfunctions.
«(*) f *•<<*) (38)
Upon putting / = in (37) and its derivative with respect to /, we have
ft = Q cos a<
gi = Qoj . sin a, (39)
The solution for other end conditions can be obtained in the same manner.
The reader should try his hand at verifying the following results:
(i) Bars with hinged ends.
r(0) = r"(0) = r (L) = and /'(L) =
Frequency equation and eigenvalues:
sin PL = (3L = n, 2tt, 3tt, • • •
Eigenfunctions :
r(x) = sin fix
(ii) iters with free ends.
r"(0) = r'"(0) = r"(L) = and r'"(L) =
Frequency equation and eigenvalues:
cos fiL cosh pL = 1
0L = 0, 4.730, 7.853, 10.996, 14.137, 17.279
Eigenfunctions:
r(x) = sinh fix + sin fix + y(cosh fix + cos /fa)
sin /5L — sinh /3L
cosh PL — cos /?L
(iii) Bars with fixed or builtin ends.
r (0) = r'(0) = r(L) = and r'(L) =
258 THEORY OF MECHANICAL VIBRATION (4.3)
Frequency equation and eigenvalues: Same as (ii), except that
pL = has no physical significance.
Eigenfunctions:
r(x) = sinh fa — sin fa + y(cosh fa — cos fa)
sinh fiL — sin fiL
cos f$L — cosh f>L
(iv) Bars with one builtin end and one hinged end.
r(0) = r'(0) = r(L) = and r"(L) =
Frequency equation and eigenvalues:
tan PL = tanh fiL
pL = 3.927, 7.069, 10.210, 13.352, 16.493
Eigenfunctions :
r(x) = sinh fa — sin fa + y(cosh fa — cos /fa)
sinh PL — sin ^L
V =
cos /?L — cosh /?L
(C) FREE VIBRATION OF A STRETCHED MEMBRANE
The differential equation of motion is given by (13) to be
a* s _ 2
= — V w
d*t P h
To separate the timevariable, we let
w = pit)  r(x, y)
= C cos (tor — <x)r(x, y)
and obtain the partial differential equation for the eigenfunction
VV + p 2 r = (40)
where
D9 P n0 >'
s
The original boundary condition w = becomes r = for (40).
(41)
(4.3)
VIBRATION OF ELASTIC BODIES
259
This time, after t is separated, we are still left with a partial differential
equation containing two spacevariables. Except for numerical methods,
there is no general way of solving a partial differential equation. Only
when the boundary conditions are simple can the solution be expressed in
terms of familiar functions — trigonometrical functions, exponential
functions, or Bessel functions. 13 But whether or not a solution can be
expressed simply, it has certain analytical properties. These we like to
r (x, y)
///////
I I l M 1 I
Figure 89
emphasize. To give an example of a solution to the boundaryvalue
problem
with
r = 0on the boundary
let us consider the case of a rectangular membrane bounded by sides
(Fig. 89).
x = y = x = a and y = b
For this problem we may again use the method of separation of the
variable by trying to find a solution of the type
r(x, y) = X(x) • Y(y)
Following a procedure similar to that used to separate t, we obtain
mnx mry
r(x, y) = sin sin — —
a b
13 This is understandable. After all, nature is capable of infinite varieties, whereas
we have only a limited number of familiar functions.
260 THEORY OF MECHANICAL VIBRATION (4.4)
and
(m\ 2 ln\ 2 B 2
in which m and n are positive integers. The eigenvalues for the problem
are therefore all values of p that can be obtained from (42) by using
integers for m and n. The complete solution is then the double series
^ ^ mrrx firry
w= 2 2 C TOW cos (co mn /  a) sin sin
ni=l«=l a (2
where, according to (41) and (42)
In general, the /5's, the eigenvalues, and the co's, the natural frequencies,
are distinct for distinct pairs of integers m and n. But if a and b are
commeasurable, that is, if a/b is a rational number, then there are some
repeated eigenvalues, For instance, if a\b = f , o> 34 = co 62 , oj 36 = co 92 , etc.
4.4 Orthogonal Property of Eigenfunctions
We have seen in Art. 4.3 that in seeking a particular solution to a free
vibration problem it is necessary to expand a given function that describes
an initial condition into an infinite series of eigenfunctions. For this
expansion process to be carried out in a simple way it is desirable that the
eigenfunctions form an orthogonal set. For instance, it can be shown that
the eigenfunctions of a vibrating cantilever beam obtained in Art. 4.3(b)
satisfy
r irj dx = i=^j (43)
Jo
where r(x) is defined by (36). With this established, the coefficients f t and
g { in (38) and (39) can be obtained by the wellknown procedure for the
determination of the coefficients in a Fourier series. The principle is as
follows: if f(x) can be expanded into
A*) = X//X*)
3 = 1
then
rHx)f(x) = fuixy^x)
3=1
(4.4) VIBRATION OF ELASTIC BODIES 261
By integrating both sides over the interval to L and utilizing (43), we have
\ L f(x)r i {x)dx=f i \\r i {x)Tdx
Jo Jo
In this way/i is found. By finding g { in the same way, the coefficient C t
and the angle oc t  in (37) can be determined through (39).
On the other hand, the eigenfunctions in Art. 4.3(a) are not orthogonal
in the sense of (43), except when ju = or oo. Thus the same procedure
cannot be used without alteration to find the constants in (25) and (26).
The treatment of this case is taken up later.
Although it is possible to investigate the orthogonality of the eigen
functions for each case, it is much better to begin with a general discussion.
For this purpose let us return to Chapter 3, where we were dealing with
eigenvalue problems of the matrix equation :
Cr = lAr
We showed in Art. 3.5(b) that the eigenvectors have the property that
?<*> C r<>> =
when X t ^ A ;
p«> A r<>) =0
In the special case in which A or C is a scalar matrix we have [see Art.
3.12(b)]
fW r (i)  o
Here we are dealing with differential equations of the form
L(r) = AM(r) (44)
The corresponding orthogonal relations to be investigated are
L
I
(45)
when X t ^ A,
r,M(r,) da =
s
where the integration is over the domain 2 occupied by the elastic body.
Note that the orthogonal relation for vectors involves a finite summation
process, whereas for functions it involves an integration process.
Let us recall that in Art. 3.5(b) the orthogonal relation for eigenvectors
is established upon the facts that the eigenvalues are distinct and that for
any two vectors u and v the symmetry of the matrices A and C leads to
u (Av) = v (A u)
and
u (C v) = v (C u)
262 THEORY OF MECHANICAL VIBRATION (4.4)
For our present problem we have a similar situation. Two eigenfunctions
having different eigenvalues will satisfy (45) if the differential operators
L and M, together with the boundary conditions, are such that for any two
functions u and v satisfying the boundary conditions and certain continuity
and differentiability requirements the following relationships hold:
u\(v) da = vL(u) da
E * (46)
j uM(v)da= j vM(u)da
In mathematical terminology we call a boundaryvalue problem self
adjoint when the foregoing relations are satisfied. To prove that the eigen
functions of a selfadjoint boundaryvalue problem satisfy the orthogonal
relations, as defined by (45), we need only to follow the same steps dis
played in Art. 3.5(b). Since eigenfunctions satisfy (44),
lr, = A,Mr,
or
r 3 Lr z = X i r j \Ar i
and
Lr, = AjMrj
or
r i^ r j = h r i^ r 3
Hence
J OM  r i lr i) da = I (ViMr,
x/jMr,) da
Because of (46), the lefthand side is zero and the righthand side can be
simplified by factoring. The result is
= (A,  X t ) I r { Mr, da
If X t ^ X j whenever i ^j,
r 2 Mr, da = r,Mr ? da —
and
r^Vj da = r,Lr 2 da =
The validity of (46) depends upon the nature of the operators L and M.
as well as the boundary conditions of the problem. In the remaining part
of this article we shall examine the existence of orthogonality among the
eigenfunctions for the systems analyzed in Art. 4.3 by investigating the
validity of (46) when applied to the differential operators in these problems.
(4.4) VIBRATION OF ELASTIC BODIES 263
The differential equations studied in (a) and (b) of Art. 4.3 can be put
into the general form of
dx n \ dx n J
where (47)
m{x) — p(x)A(x)
When n = 1, this equation governs the eigenfunction for longitudinal
vibrations of bars, lateral vibrations of a stretched string, and torsional
vibrations of shafts. When n = 2, it governs lateral vibrations of slender
bars. The parameters k and m represent the stiffness distribution and the
mass distribution, respectively, along the bar, the shaft, or the beam. In
general, these parameters are functions of x, but frequently they are
constants. In the latter case the equation simplifies into
^r=(l)"/? 2 V (48)
where
mw 2
R2n
Referring to our previous notation, we see that the two differential
operators L and M are
d n
dx n
. dx n .
= D n [k(x)D n ] (49)
and
M = m(x)
where D stands for djdx.
Since the operation by M is merely a multiplication by a function m(x),
it is obvious that the second of the two equations in (46) is automatically
satisfied. We must concentrate now only on establishing the first equation.
Let us take the boundary of the system to be at x = and x = L.
Casel. L = D(A:D)
For this case (47) becomes
dx \ dx,
Or more simply
— I A: — = —mcrr
dr. \ dx]
(kr')' = —mo 2 r
264 THEORY OF MECHANICAL VIBRATION (4.4)
The group of vibrational phenomena governed by this equation is discussed
in Art. 4.2(a). The difference between the two sides of the equations in
(46) is then
f [ul(v)  vl(u)] dx = f [u(kv'y  v(ku') f ] dx
Jo Jo
Through integration by parts, we obtain
[ul(v)  vl(u)] dx = [ukv'  vku'}% (50)
.
o
If the boundary conditions are such that any two continuous and differen
tiate functions u and v satisfying them will make the righthand side of
(50) vanish, then the problem is selfadjoint, and its eigenfunctions are
mutually orthogonal in the sense of (45), provided that the eigenvalues are
distinct. In other words, if
[r&rf)  r/fc//)tf = (51)
it follows that
**j
and
f VW dx = [ L ri (x)[k(x) r ;(x)\ dx =
Jo Jo
I r t M(r,) dx = mixy&y/x) dx = i^j (52)
Jo Jo
The second of the two conditions is the one we need for the expansion of a
given function into an infinite series of eigenfunctions.
Let us now examine the boundary conditions for which (51) is valid.
It can be shown by simple substitution that if the two boundary conditions
are both of the linear homogeneous form
ar + br' = (53)
then (51) is always satisfied. In (53) the coefficients a and b may have
different values at different boundary points but they must not depend on
the eigenvalues of the problem. Take the case of longitudinal vibrations
of bars as an illustration. The general form (53) includes such common
end conditions as
(i) fixed end r =
(ii) free end r' —
(iii) end terminating in an elastic spring, K,
OX
or
K
r r
EA
(4.4) VIBRATION OF ELASTIC BODIES 265
When the condition at each end of a bar belongs to one of the three
kinds described, the eigenfunctions are orthogonal to one another with
respect to the weighting function m(x).
m{x)rjx)rj(x) dx = i^j (54)
Jo
The weighting function m(x) is the mass distribution along the bar
m(x) = p(x)A{x)
For homogeneous bars with uniform cross sections m is a constant, and
we have the ordinary orthogonal relation
\ L r i (x)r j (x)dx = i^j (55)
Jo
Let the expansion of a given function /(#) into a series of eigenfunctions
satisfying (54) be
/(*) = £/w«)
i=i
The coefficients /can be found by
Jm(x)r i (x)f(x) dx = f t m(x)[r i (x)] 2 dx (56)
o Jo
or
m(x)r^x)f(x
Jo
)dx
m(x
Jo
)[rlx)fdx
Let us now return to problem of Art. 4.3(a), in which one end of the bar
terminates in a concentrated mass M. The boundary condition at such
an end is given by (22) as
E
Even though this boundary condition is linear and homogeneous, it does
not belong to the type described by (53) because the second coefficient
contains o>, which is not the same for all eigenfunctions. Therefore, the
eigenfunctions in this problem sin f$ t x, in which p { are the roots of (24),
do not satisfy (55). In other words,
CL
sin ftp sin fife dx ^
Jo
266 THEORY OF MECHANICAL VIBRATION (4.4)
This lack of orthogonality not only causes difficulty in the finding of the
coefficients f t and g t in (26) but also invalidates a number of important
analytical results based on the orthogonality or eigenfunctions. Fortu
nately, by interpreting the problem in a different way, the desired property
of orthogonality can be preserved. Let us observe that in the initial
formulation of the problem a bar of uniform mass distribution is the
vibrating elastic body under consideration, whereas the existence of an
attached concentrated mass M enters into the problem only as a boundary
E,A,p,I
Figure 90
condition. The mass distribution of the system is therefore considered
uniform as given by
m(x) = pA = constant
We can, however, reformulate the problem by considering the concen
trated mass as a part of the vibrating bar, which will then have a free end.
Suppose at first we redistribute the concentrated mass uniformly along a
small portion of the bar near the right end of length e. The right end is
then free (Fig. 90). Let the eigenfunctions for this mass distribution be
denoted by
Me,x) i=l,2, •••
in which e serves as a parameter. By physical reasoning,
lim 0,(€, x) = sin p. ( x
since a concentrated mass is the limiting case of a finite mass distributed
over an infinitesimal region. In the meantime the eigenfunctions are
orthogonal with respect to the modified mass distribution function mfc, x)
where
w(€, xyt>l€, *)<£/€, x) dx = i+j (58)
m(e, x) = pA < x < L — e
m(e, x) == pA + jupAL/e (L — e) < x < L
(4.4) VIBRATION OF ELASTIC BODIES 267
Because of (58), we can apply (57) and take the limit as e >0. Consider
the numerator of (57) first.
m(e, #)</>;(€, x)f(x) dx
Jo
L
 P A
^ V/€, X)/(X) dx+(\+ ^U _ &(*, *)/(*) <fc
For sufficiently small e the integrand in the second integral must be
continuous. Hence we may use meanvalue theorem to obtain
Jm(e, ^)</> z (e, x)f(x) dx
o
= p a\ J V*fo *)/(*) * + (€■+ ^W*. f)/tt)
where L — e < £ < L.
Hence 14
lim
e^OJO
m(e, x)^^, x) f (x) dx
In
= p/4 J sin PfX f(x) dx + //L sin p t Lf(L)
_Jo
Similarly
f
lim m(c, #)[<^(e, x)] 2 dx
e>oJo
= p^ I sin 2 &# cte + juL sin 2 /3,L 1
sin pixfix) dx + //L sin ^Lf(L)
Jo
The result is
J sin 2 j8 t x afc + /uL sin 2 &L
o
Although the formula just derived is for a concentrated mass at one end of
a uniform bar, the principle behind the derivation is applicable whenever
concentrated masses produce eigenfunctions not orthogonal in the ordi
nary sense. Mathematically speaking, the mass distribution function m(x)
in the general formulation of the problem described by (2) is allowed to
behave like Dirac's ^function, with singularities at a finite number of
points, yet bounded elsewhere and integrable over the domain occupied
14 We are using a convergence theorem, attributed to Lesbeque, which permits the
passing of the limit sign from the outside to the inside of the integral.
268 THEORY OF MECHANICAL VIBRATION (4.4)
by the system. When the existence of concentrated masses is treated in
this way the eigenfunctions are again mutually orthogonal with respect to
this modified density function.
Another broad implication of the example studied is that, although in a
differential equation formulation of a dynamics problem we may be forced
to define the boundary of our systems in such a way that no concentrated
forces may appear at the interior points in formulations by integral
equations or energy equations, such restrictions may not be needed. This
aspect of the problem is to be amplified later.
Case 2. L= D 2 (kD 2 )
Having studied the case of n = 1 in (49), we can now follow the same
procedure for n = 2. This case pertains to the lateral vibration of bars or
beams. The eigenvalue problem is embodied in the differential equation
■(*£')
d 2
or
(kr")" = moj 2 r
where k(x) = EI = bending stiffness of cross sections
m(x) = pA = mass distribution along the bar
The parameters E, /, A, and p may be constants or variables. The next
thing is to investigate the conditions under which the first equation of (46)
is valid. The second one is obviously valid. Again, through integration
by parts, we have
Pi/L(i?) dx = \ L u(kv"y dx = [u(kv")']%  f L u '(kv")' dx
J Jo Jo
= [u(kv")'  u\kv'X + f u'kv" dx (59)
Jo
Similarly,
I vl(u) dx = [v(ku"Y  v\ku")]L + I v"ku" dx
Jo Jo
Since the integrals at the righthand sides of the two equations are the
same, the difference between the lefthand sides is
[ul(v)  vl(u)] dx = [u(kv'J  v(ku"Y  u'kv" + v'ku'% (60)
Again the boundary conditions of the problem will determine whether it is
selfadjoint and its eigenfunctions, mutually orthogonal. The physical
meaning of the terms at the righthand side of (60) should now be reviewed.
(4.4) VIBRATION OF ELASTIC BODIES 269
It is seen that
u, v = deflections at the ends
u', v — slopes at the ends
ku ', kv" = bending moments at the ends
(ku")' ', (kv") = shear forces at the ends
Therefore, for any of the following simple end conditions each of the four
terms at the righthand side of (60) vanishes individually. (See Fig. 86.)
At builtin ends r = r =
At pinned ends r = kr" = (61)
At free ends kr" = {kr")' =
A more general way whereby (60) will vanish is that each end condition
is a linear homogeneous relationship either between the deflection and the
shear or between the slope and moment. Analytically, it means that at
each end there is a set of constants a, b, c, and d such that the boundary
conditions may be expressed by
ar + b(kr"Y =
and (62)
cr' + d(kr") =
The conditions stated by (60 are special cases of (62). The reader can
verify that if u and v satisfy (62), then (60) vanishes. Physically, in the
most general case (62) represents the end conditions when the ends are
elastically supported. The stiffness of the support may be zero or infinite.
Sometimes a bar terminates in a concentrated mass. At such an end
the linear relationship between the deflection and shear depends upon
frequency. The orthogonal relation among the eigenfunctions will then
have to be modified in the manner already discussed. The result is
1
L
m(x)r i (x)r j (x) dx + Mr^Qr^L) = i ^ j
where M is the concentrated mass at x = L.
Case 3. L = V 2
The twodimensional wave equation (13), after separation of time
variable f, reduces into
V 2 r = —o 2 kr
where
k(x, y) = phjS
270 THEORY OF MECHANICAL VIBRATION (4.4)
The integral to be examined this time is
I
(u V 2 v  v V 2 w) dA (63)
This integral is over a twodimensional region A. Our previous technique of
integration by parts cannot be used here, but there is a procedure that is
entirely equivalent. This procedure is best described with notations in
vector analysis. It is known that the Laplacian operator V 2 is the diver
gence of a gradient
V 2 = div (grad) = V • V
Hence
u ^v = wV • (Vv) = V • [u(Vv)]  (Vw) • (Vr)
Similarly,
v V 2 w = V • [v(Vu)]  (Vr) • (Vt/)
Substituting into (63), we have
f (u V 2 v  v V 2 w) dA= f V • [w(Vr)  v(Vu)] dA
Now we apply divergence theorem to the righthand side and obtain the
statement of Green's theorem.
£
(u V 2 v r v V 2 w) dA = (b [u(Vv)  r(Vw)] ■ ds
The righthand side is a contour integral around C, which is the boundary
of A. It can also be written as
I (u V 2 v  v V 2 w) dA = <j> (u^  v^\ ds (64)
in which djdn stands for "the normal derivative of" at the boundary.
Hence, if the boundary condition of the problem is either
r = on C
or
^ = 0onC
on
then (64) vanishes and the eigenfunctions with distinct eigenvalues are
orthogonal in the sense that
n
k(%, y)ri( x , yVjix, y) dx dy = (65)
(O t ^ COj
(4.5)
VIBRATION OF ELASTIC BODIES
271
The special case of two different eigenfunctions with the same eigen
value can be handled by a principle similar to that used in Art. 3.10.
If r t and r j are two independent eigenfunctions with the same eigenvalue
and they are not orthogonal to each other in the sense of (65), it is possible
to replace one of the two functions by a certain linear combination of the
two functions, which is also an eigenfunction and satisfies (65). We shall
not go into the details of finding this combination. After all, the situation
is highly pathological because a very small change in the system will
separate the eigenvalues and result in two orthogonal eigenfunctions.
Case 4. L = V 2 (&V 2 )
Having gone through three similar cases and a general discussion, we
shall simply give a key derivation for this case. The reader can easily
fill in the rest of the details if he wishes.
By formulas in vector analysis
and
Hence
uV 2 (k V 2 v) = V • [uV(k V 2 r)]  Vw • V(k V 2 r)
Vw • V(k V 2 r) = V • [k V 2 v(Vu)]  k V 2 r V 2 «
f [uV 2 (k Vh)  vV 2 (k V 2 w)] dA
= ( V • [uV{k V 2 v)  k V 2 r(Vw)  vV(k V 2 u) + k V 2 w(Vr)] dA
u — (k V 2 iO  k V 2 r^ v — (k V 2 w) + k V 2 u ~
on on on on
ds
The problem is selfadjoint if this integral around the boundary vanishes.
4.5 Formulation by Integral Equation
In Chapter 3 we showed that equations of motion can be written with
either the elastic constants or the influence coefficients for positive
definite systems. The matrix operators formed by the influence coefficient
is the inverse of that formed by the elastic constants, and vice versa.
Hence from
C r = oj 2 A r
we can obtain the equations by influence coefficients with an inverse
operation.
C i Ar
or
272
THEORY OF MECHANICAL VIBRATION
(4.5)
We expect that similar inverse operations may exist for the differential
operators being studied. That is, for
Lr
Mr
(66)
there is an equivalent relation written as
L 1 Mr = I  r
or
Since L is a differential operator, L _1 must denote an operation involving
an integration process. To find the nature of this inverse operation, we
have chosen to approach it on the basis of the physics of the problem
rather than to discuss it as a purely mathematical theorem. (See Exercise
4.16.)
Unit force
Figure 91
Consider for example the lateral vibration of a bar supported in some
way. (Fig. 91). From the way the bar is supported (boundary conditions)
and from its dimensions and makeup (operator L) we can find the in
fluence coefficient between any pair of interior points in the bar. The
influence coefficient for a given system is therefore a function of two
points, which in the case of a slender bar are designated by two variables,
say, the distances of the two points from the left end, x and . In mathe
matics this function is called Green s function and is denoted by the symbol
G{x, £). For linear systems Maxwell's reciprocal theorem states that
G(x, ) = Gtf, x)
or that Green's function for a linear system is symmetrical.
Let us now assume that the bar is vibrating in one of its natural modes.
The displacement along the bar is then described by
v(x, t) = Cr(x) cos ((ot — a)
(4.5) VIBRATION OF ELASTIC BODIES 273
The inertia forces acting on the bar have the distribution
d 2 v
— m(x) — — = Cco 2 r(x) cos (cot — a) • m(x) (67)
or
where
m(x) = p(x)A(x)
The relation between the inertia forces on the bar and the displacement at a
generic section x is then
v(x, t)
f
Jo
G(x,£)m(£) — v(
£, t) d£
Or, after elimination of
cos (cot
— a),
f
G(x, £);
wtfMO di =  2
or
r(x)
Symbolically,
this equat
ion can
L
be expressed by
L (Mr) = i 2 r
CO 1
(68)
(69)
where M = m, and L _1 represents an integration operation.
The integral equation (68) is called a linear homogeneous integral
equation of the second kind. It states a typical eigenvalue problem : "Given
an operation characterized by G and m, find a function r which remains
unchanged by this operation except for a multiplication factor." Stated
in another way the problem is to find the values of co 2 for which (68) can
have nontrivial solution, the trivial solution being
r(x) =
Let us note the fact that in this formulation no boundary conditions are
mentioned. This is because the information concerning boundary con
ditions is already included in Green's function of the problem G. It is a
simple matter to show that if G(x, £) satisfies a linear homogeneous
boundary condition
G + aG' + bG" • • • = (70)
in which the prime denotes differentiation with respect to x, then any
function r satisfying (68) must also satisfy the same boundary condition.
It should also be borne in mind that unless the boundary condition is
linear homogeneous the superposition principle on which the formulation
is based will no longer be valid.
274 THEORY OF MECHANICAL VIBRATION (4.6)
Although the lateral vibration of bars was chosen initially for this
discussion, the formulation presented is quite general. The integral
equation (68) represents all linear eigenvalue problems having one space
variable. For multidimensional problems (68) is generalized into
^G(P,Q)r(Q)da Q = ^ 2 r(P) (71)
j:
in which P and Q represent the coordinates of two generic points in the
elastic body. If the displacement r is not unidirectional, we shall have
more than one equation of the form (71).
The usual analytical method of solving an integral equation, such as
(68), is to find the associated differential equation, such as (66), and solve
it. In this way, there is no advantage in formulating a problem by integral
equation as long as we can always formulate the same physical problems
by differential equations. However, the integral equation formulation has
great heuristical value in numerical analysis. For instance, the form
L 1 Mr = — /•
or
immediately suggests an iteration procedure for the determination of the
eigenfunction with the lowest co 2 . The scheme will be the same as that
described in Art. 3.11(b). The numerical method for solving this integral
equation is discussed in Art. 4.11.
4.6 Rayleigh's Quotient and Its Stationary Values
(a) rayleigh's quotient for free vibration of an elastic body 15
In Chapter 3 we showed that the eigenvalue problem of the matrix
equation
C r = ( o 2 A r
is related to the finding of stationary values of the associated Rayleigh's
quotient
u C u
Q u = — 
u A u
Briefly we found that
(0 Q( u ) is greater or equal to the lowest eigenvalue. Or
<2(u) > co 2 if co 2 < co 2 \ • ■ • , < co*
15 To gain a better understanding of this discussion the reader is advised to review
Arts. 2.6 and 3.6.
(4.6) VIBRATION OF ELASTIC BODIES 275
(ii) If u is an eigenvector, Q(u) is the associated eigenvalue.
e(<" (i ») = co, 2
(iii) If u is only slightly different from a modal vector, then Rayleigh's
quotient of u differs from the corresponding eigenvalue by a secondorder
small amount. In other words, Rayleigh's quotient achieves a stationary
value when u is one of the modal vectors.
It is also important to remember that one essential premise for these
conclusions is that
u A u > u C u >
The analysis in Art. 3.6 suggests heuristically that for the eigenvalue
problem in this chapter
L(r) = w 2 M(r)
We may form Rayleigh's quotient of a function u(x, y, z)
uL(u) da
Q(fu)) = ^ (72)
u M(w) da
which perhaps will have similar properties.
This time the value of the quotient Q depends on the form of a function
u instead of a vector u, hence it is a "function of a function" or afunctional,
denoted here by a double parenthesis as used in (72).
Having noted the similarity we shall now mention the obvious difference
between the two cases. In forming Q(u), the vector u is completely arbi
trary, whereas the function u in Q((u)) is subjected to some restriction.
In the first place, the function u must be sufficiently regular in its continuity
and differentiability so that L(w) and M(w) can be defined and the two
integrals in (72) have meanings. In the second place, since the boundary
conditions of the problem as well as the differential equation describe
the physical nature of the vibrating system, these conditions must somehow
enter into consideration.
Let us consider a class of functions called admissible functions. 1 * which
satisfy all the boundary conditions of an eigenvalue problem and for which
L(w) and M(w) are continuous functions inside 2. Obviously, the eigen
functions themselves must belong to this class. Any function u belonging
to this class can be expanded into an infinite series of eigenfunctions,
16 There is no universally accepted meaning for this term. It is thus necessary to
examine the conditions of admissibility whenever one sees the term used.
276
THEORY OF MECHANICAL VIBRATION
(4.6)
which can be differentiated termwise to yield series representing L(V) and
M(w); and all of the series thus obtained are uniformly convergent.
Accepting this statement without proof, we may write
u = v x r x + v 2 r 2 \ = 2 v t i
(73)
i = l
where the coefficients v i for selfadjoint problems are evaluated by methods
in Art. 4.4. By termwise differentiation, we have
L(") = I>M = 5>,VM(/v)
1 = 1
= 1
and
M(:
") = m(^) =^M(.,)
(74)
(75)
Let us assume that the boundary conditions are such that the problem
is selfadjoint. The eigenfunctions are then mutually orthogonal with
respect to both L and M. This orthogonality, together with the uniform
convergence of (73) and (74), enables us to write
f* co co /•
wL(w) (/ff = n ViVjCofcMir,) da
i = l JS
Similarly, 17
I uM(u) da = f v? I ^M(r,) tfo
For simplicity let us write
m it ^jr t M(r t )da (76)
Rayleigh's quotient (72) for an admissible function u can thus be written
G(M) =
Wu«iV + m 22 v 2 2 co 2 2 +
W + ^22^2 + * '
CO
2 m,^, 2 co, 2
co
2 "»«»<*
(77)
17 If (74) is uniformly convergent, so (75) will be.
(4.6) VIBRATION OF ELASTIC BODIES 277
Except for its infinite character, this expression resembles (39) of Art. 3.6
in every other way. Inasmuch as both the numerator and the denominator
are convergent series, we can use the same reasoning employed in Art. 3.6
and conclude the following:
(i) Rayleigh's quotient for any admissible function must be at least
equal to the lowest eigenvalue of the system. (This conclusion is contin
gent upon the fact that m H is nonnegative, a fact which will become clear
later. See also Exercise 3.11.) To show this, we need only to observe that
(77) can be written
__ 2 , m 22 v 2 \w 2 2  «)*) + m 3S v s 2 (oj 3 2  to 2 ) • • •
Q((u)) = o h * +
2 m u v i 2
i = l
(ii) If u = r it from the orthogonal property of eigenfunctions
v ± = v 2 = • • • = v^ = v i+1 = • • • =
Hence
fi(W) = <
(iii) In the "neighborhood" of an eigenfunction Rayleigh's quotient is
stationary. That is to say, if an admissible function is only slightly different
from an eigenfunction, its Rayleigh's quotient differs from an eigenvalue by
a "higherorder" small amount. Although this statement is not mathe
matically precise, it is intuitively useful. To justify this statement, we
observe that since every admissible function u is supposed to have an
infiniteseries expansion consisting of eigenfunctions whose coefficients
determine collectively the original function we may consider Q{{uj) as a
function of these coefficients as given by (76):
Q((u)) = Q(v v v 2 , • • • , r, • • • )
A small change in the function u corresponds to a set of small changes in
the coefficients i\, v 2 , ■ • • , v i9 .... The corresponding change in Q,
denoted by the symbol 6Q and called the variation of Q, is given by
dQ = — di\ + 3— 6v 2 + • • •
di\ dv 2
= 1 ^— d»i
i = l OVi
Accepting the fact that here we can treat a function of (countable) infinite
number of variables by the usual rules of calculus, we may easily show
dQ
— — = when u = r ;
dv,
278 THEORY OF MECHANICAL VIBRATION (4.6j
for all i and/ By u = r j9 we mean that
Vl = v 2 = • • • = y^ = p m = • • • =
and
In other words
(3Q =
when u is one of the eigenfunctions, say r jm
(b) minimum characterization of eigenvalues
A better and more precise way of characterizing the stationary property
of Rayleigh's quotient is as follows. Arrange the eigenfunctions in a
sequence with ascending eigenvalues:
h, r 2 , • • • , /•„ • • •
oh 2 < oj 2 2 <,' — , co?, • • •
Suppose that we remove from the class of admissible functions all those
which are not orthogonal to the first (k — 1) eigenfunctions of the system.
In other words, we select from the original class of admissible functions a
subclass in which all the functions are orthogonal to r v r 2 , . . . , r^. In
this smaller class the function that has the minimum Q is r k . Rayleigh's
quotient of any other functions in this subclass will be greater than m k 2 .
The reader is left with the proof of this statement. In proving it one needs
only (77) and the orthogonal relation of eigenfunctions.
(C) MAXIMUMMINIMUM CHARACTERIZATION OF EIGENVALUES
Let
</> l5 25 • • • , (/> k _ 1
be a set of (k — 1) functions, which are completely arbitrary except that
they are integrable in S. We remove from the class of admissible functions
all those which do not satisfy the relation 18
J </> ; M(w) da = j = 1, 2, • • • , k  1
(78)
In other words, we retain only those admissible functions satisfying (78).
Again we end up with a smaller subclass. This subclass will be the same
as that in (b) if the </>'s are the first (k — 1) eigenfunctions; otherwise they
18 This relation will mean orthogonality between j> } and // if the operator M is such that
(46) is satisfied.
(4.7) VIBRATION OF ELASTIC BODIES 279
will be different. We now have the following theorem: the minimum
value of Q formed by functions in this subclass is not greater than the
kth eigenvalue, or
[fi((«))Li„ < ^
in which u is an admissible function satisfying (78). To prove this theorem,
we show first that there is a function u belonging to this subclass, which
can be represented by a linear combination of the first k eigenfunctions:
"o = *Vi + v*r 2 , • • • , v k r k (79)
where v l9 v 2 , . . . , v k are the coefficients. If there is a w , then according
to (77)
n(( xx _ "%g>iV + m 22 a) 2 2 v 2 2 + • • • + m kk oj k 2 v k 2 2
m lx v 2 + m 22 v 2 2 + ' ' • + m kk v 2
To prove that indeed there is such a function we must show that it is
possible to find v lf v 2 , . . . , v k for (79) so that (78) will be valid. It is not
difficult to show that by substituting (79) into (78) we can obtain (k — 1)
linear homogeneous equations with k unknowns, namely, v l9 v 2 , . . . , v k .
where
fllVl +/l2^2 + ' ' ' +flk V k = °
fttPl +/ 2 2^2 + • • ' +/ 2 fc^ = °
f(k 1)1 V 1 + J(kl)2 V 2' ' ' if{kl)k V k = ^
(80)
f H = J 1 ^Mir^da (81)
j= l,2,,(/c 1)
i = 1,2, • • • ,k
This set of equations (80), having more unknowns than equations,
evidently has a solution, though not a unique solution. This completes the
proof.
4.7 RayleighRitz Method
(a) mathematical principle
The discussion in Art. 4.6 reveals that if we have a way of finding in a
class of functions the one that gives the least value for Q we shall either
have the solution to the eigenvalue problem or establish the lower or
280 THEORY OF MECHANICAL VIBRATION (4.7)
upper bound of the eigenvalues. To perform this minimization process to
a class of all admissible functions is equivalent to solving the original
problem in differential equation, which cannot always be carried out. In
the RayleighRitz Method we apply the minimization process to a smaller
class of admissible functions and thereby obtain an approximate solution
to the problem.
We begin by selecting a set of n linearly independent admissible functions :
Except for the requirement of linear independence and the condition of
admissibility previously defined, the selection of these functions can be
done in an arbitrary manner. From this set of </>'s and by taking linear
combinations of (/>'s, we can form a class of functions, which have the
general form
u = ^ + q 2 cf> 2 , • • • + q n (f> n (82)
in which the g's are a set of arbitrary coefficients. It is obvious that every
function belonging to this class is an admissible function but not all
admissible functions are in this class. The set of functions </> is called a
generating set. It generates a class of functions u defined by (82).
Let us define
■i da
(83)
D, da
For selfadjoint problems
a u = a it and c u = c H
Rayleigh's quotient for any function u representable by (82) can be
obtained by substituting (82) and (83) into (72). The result is
it n
2 2 c a<ii<ii
Q((u)) = t=li=l
2 2 a a a i a J
Or, by the summation convention used in Chapter 3, we have simply
CjMl _ qCq
(4.7) VIBRATION OF ELASTIC BODIES 281
In this expression a tj and c ij are predetermined by the initial choice of
<f>, the generating set. The q's, however, are still free to vary. The amount
of freedom is thus equivalent to that of «degreefreedom systems. To
minimize Q, with respect to all possible combinations of q's, we set
dQ dQ dQ
dq 1 dq 2 dq n
and Q = & (86)
The procedure for combining (84), (85), and (86) is exactly the same as that
leading from (26) to (31) in Art. 2.6. The result is the familiar equation
K,co 2  c i} \ = (87)
In general, this equation will have n roots: d^ 2 , co 2 2 , . . . , co n 2 , arranged
in the ascending order of magnitude. To each of these roots there associates
a modal vector (q l9 q 2 , . . . , q n ), which in turn yields a function r through
(82): '««,«,
r = Wl + ?2<?2 + * * ' + qj>n
In total we have n such functions, r l9 r 2 , . . . , r n .
The essence of this method is that by limiting the admissible functions
to a class representable by (82), we are approximating a system having an
infinite number of degrees of freedom by one having n degrees of freedom.
The n frequencies and modal vectors of the latter are then considered as
giving an approximate description of the first n modes of vibration of the
original system. To amplify this statement, let us observe that r x is the
function with the smallest Q in the class of functions represented by (82).
If this class is "sufficiently representative" of the original admissible class,
r ± is an approximation of r l9 which is characterized by having the smallest
Q of all admissible functions. In the meantime cb^ 2 is a better approxi
mation to o)^ than /\ is to r x because of the stationary property of Q. In
any event, r 1 and w x 2 are the best approximations available within the free
dom of choice allowed by (82).
There is, of course, no guarantee that the approximation will be good.
For instance, it can conceivably happen that all of the <£'s in the generating
set will be orthogonal to r v In that case, r x must also be orthogonal to r x
and can never be considered as an approximation. In practice, this does
not happen because we generally have some idea of what r x should be like
and choose the generating set accordingly. Now in the approximating
system it can be verified readily that r 2 is orthogonal to /\ with respect
to M.
b
Mr 2 da =
282 THEORY OF MECHANICAL VIBRATION (4.7)
Hence r 2 is the function with the smallest Rayleigh's quotient among all
functions orthogonal to f x and representable by (82). This description
parallels the minimum characterization of r 2 , except that there we consider
a larger class of admissible functions. It is clear then that, if ^ were
exactly r l9 co 2 and r 2 would be only approximations for oj 2 and r 2 , respec
tively. Since r x is only an approximation for r l9 oj 2 and f 2 in general can be
only rougher approximations of oj 2 and r 2 . Similar things can be said for
the rest of eigenvalueeigenfunction pairs.
Because we use a smaller class of admissible functions in the Rayleigh
Ritz method we know immediately that
to 2 = R((rj) < RdrJ) = o\ 2
It is also true that
co 9 2 < tb 9 2 , co J 1 < co A • • • co 2 < m.
but the proof of these inequalities is not within our reach at the moment.
To avoid excessive digression we shall put the matter aside and return to
it later in Art. 4.8(b).
(b) liberalization of requirements for admissibility by
energy consideration — an example
Up till now an admissible function u eligible for consideration in the
minimization process had to satisfy all the boundary conditions of the
problem. Besides, L(w) and M(w) had to be defined and to be continuous
everywhere. Both of these requirements can be liberalized. For instance,
the continuity requirement for L(w) and M(w) can certainly be relaxed
somewhat. In other words, to be valid, we can have a larger class of
admissible functions and still have all the conclusions arrived at in Art. 4.6.
To carry out this liberalization by a general and strictly mathematical
approach is neither feasible here nor conducive to a clear understanding of
the physical problem. Therefore, this time we choose to develop the
theory by the use of an example.
Consider the problem of lateral vibrations of a bar with a builtin left
end and a right end fastened to an elastic support (Fig. 92). The separation
of timevariable t from the partial differential equation (10) results in
{kr") n = co 2 mr (88)
where
k(x) = El and m(x) = pA
The boundary conditions to be satisfied by r are
r (0) = 0, r'(0) = (89)
k(L)r"(L) = K/iL) and [k(L)r"(L)]' = A>(I) (90)
(4.7)
VIBRATION OF ELASTIC BODIES
283
in which K x and K 2 represent the rotational and the lateral stiffness of the
rightend support. Note the proper signs for the righthand side of (86)
in order for K x and K 2 to be positive.
The problem is selfadjoint, since the boundary conditions are in the
form of (62).
When k and m are constants, this eigenvalue problem can be solved
without much difficulty. But, if there are variations in the stiffness and
mass distribution along the bar, the problem is usually solved only by an
P,E,I,A
'///////A
Figure 92
approximate method based on the RayleighRitz principle. According to
our previous discussion, we shall have to choose a set of admissible
functions <j> x , </> 2 , . . . , <f> n9 which satisfy the boundary conditions (89) and
(90). Although it is not too difficult to find a set of these functions, they
may make the subsequent computation algebraically complicated. Hence
the relaxation of boundary conditions is of practical interest.
Let us suppose for the time being that we are still dealing with functions
satisfying our original requirements of admissibility. The numerator of
Rayleigh's quotient for an admissible function u is
f uL(u) da = f u(x)[k{x)u"(x)Y dx
According to (59), integration by parts yields
J ul(u) da = f k{uf dx + [u(ku"Y  u'(ku")]% (91)
Since u satisfies (89) and (90), we have finally
I ul(u) da=[ k{u"f dx + K 2 [u{L)f + K^u'iL)] 2 (92)
Jl Jo
284 THEORY OF MECHANICAL VIBRATION (4.7)
As we examine the three terms at the righthand side of (92), we see that,
except for a factor of J, the first term represents the strain energy in the bar
and the last two terms represent the strain energy in the two supporting
springs. Hence
£
ul(u) da = 2V m (93)
if u satisfies all the boundary conditions. The physical meaning of the
symbol V m is the maximum potential energy the system would have if its
vibration were described by
v(x, t) = u{x) cos (cot — a) (94)
As the denominator of Rayleigh's quotient of u, we have
uM(u) da = m{x)[u(x)f dx = — T m (95)
Js Jo or
in which T m is the maximum kinetic energy the system would have if its
vibration were described by (94). Therefore, Rayleigh's quotient can be
expressed as
co 2 V m
Q = =r* (96)
The quantity co 2 actually does not appear because it is also contained in
T m and will be canceled out when the energy expressions are written
explicitly. It was also shown in Art. 2.6 that the minimization of (96) is
equivalent to the minimization of the expression (T m — V m ); or 6Q =
implies
W m  VJ = (97)
Thus we can characterize the natural frequencies of a vibrating system as
the stationary values of the ratio in (96). This is true regardless of the
number of degrees of freedom the system may have.
Now we must realize that in deriving (96) from (91) we were actually
reasoning backward. Without going into topics in advanced dynamics,
we somehow feel that (96) is a more basic expression of Rayleigh's quotient
than (72), since, unlike (72), it contains welldefined physical quantities
and its definition does not contain explicitly such mathematical operations
as differentiation and integration. Accepting as a fundamental physical
fact that 19
• * m
19 Note also that the orthogonal property of eigenfunctions needed to arrive at (77)
also becomes unnecessary if (97) is accepted as a basic physical law.
(4.7) VIBRATION OF ELASTIC BODIES 285
with T m and V m as the maximum energies in a natural mode of vibration,
we realize that many of the requirements are imposed on the admissible
functions simply to make the lefthand sides of (93) and (95) become the
energy expressions. For instance, the function u must satisfy the boundary
condition (90), so that (93) will then be valid. But, on the other hand, we
can obtain directly through formulas in strength of material
2V m = k(u"f dx + KMP? + K 2 [u(L)f (98)
Jo
without the benefit of (92).
Looking at (98), we see now that the admissibility requirements for the
present problem are simplified :
(i) An admissible function u must be at least twice differentiate and
u" must be integrable between the limits and L. This is in contrast to the
fourtime differentiability needed in defining L(«), although this relaxation
of requirement is of little practical significance, since in picking a set of
</>'s for the RayleighRitz method we generally choose functions with
regular behavior.
(ii) An admissible function u for the problem on hand must satisfy the
two boundary conditions in (89), but not necessarily those in (90), if
Rayleigh's quotient is defined by (96) instead of (72). The difference in the
two types of boundary conditions lies with the fact that the use of potential
energy expressions automatically takes care of the conditions at the right
end elastic support, whereas they cannot take care of the fixed support at
the left end. This is because we can include the two springs in our vibrating
system. We cannot do the same for the fixed support without introducing
ambiguity in the energy expression, for whenever a boundary point is
immovably constrained the presence of an infinitely rigid and infinitely
large external body is implied. If this body is considered part of the
system, the energy expressions become indeterminate. If this body is
considered external to the system, the boundary conditions it imposes must
be obeyed by the functions eligible for consideration in minimizing (96).
Generally speaking, if
dx n \ dx nX 7
all boundary conditions containing no derivatives higher than the (n — \)th
order must always be satisfied. Such boundary conditions are thus called
imposed boundary conditions. They usually describe constraints on the
system that are purely geometrical; hence they are also called geometrical
286
THEORY OF MECHANICAL VIBRATION
(4.7)
boundary conditions. On the other hand, boundary conditions contain
ing 20 derivatives of the nth order or higher are called natural or dynamical.
These boundary conditions do not have to be satisfied by the admissible
functions in minimizing (96), since they describe dynamical conditions at
the boundary that will automatically be satisfied by the function obtained
from the minimization process. Functions satisfying all of the imposed
boundary conditions and having the differentiability described in (i) are
called essentially admissible functions. 21
Returning to our example of a bar with a builtin end and an elasticallv
supported end, we see that the following functions are examples of essen
tially admissible functions :
nirx
<p n = 1  cos
2L
4>«
.n+2
n = 1
1,2,
2,
(j> n — sinh 2
nx
1,2,
To apply the RayleighRitz method, let us choose as the generating set the
following four functions:
(7TX\ I 2)7TX\ I 5lTX\
1cosj ('cos — ) (1cos — ) and
1 — COS
21
The system is thus approximated by one having four degrees of freedom.
The functions generated by this set are representable bv
2*.
1 \ 7TX
2/T
 cos in ]
U =M n 2)L q " Sm [ n 2k
U "= X{ n ' l 2J Z" 2 ^ COS ( ,7 ^T
For the purpose of illustration let us assume
k = constant m = constant
20 It is assumed that these highorder derivatives cannot be eliminated in the set of
boundary conditions of the problem.
21 L. Collatz, a German author who writes a great deal on the subject, calls such
functions simply zulassige (admissible) functions and uses the term Vergleichsfimktionen
(compatible functions) for those satisfying all the boundary conditions. He also uses
the adjectives wesentliche (essential) and restliche (remaining) for imposed and natural
boundary conditions, respectively.
(4.8) VIBRATION OF ELASTIC BODIES 287
The strain energy due to the bending of the bar is thus
The strain energy in the two supporting springs is
= 2 K i(ii + ^ + % + ? 4 ) 2 + 2 M ^) ^ ~ 3 ^ 2 + 5 ^ 3 ~ 7 ^) 2
By putting the two together, we have a strain energy expression that is a
quadratic form in g's:
in which the repeated indices i and / are summed over the range 1 to 4.
Similarly, the maximum kinetic energy expression T m is
dx
in which d tj is the Kronecker delta (d {j = 1, when i =j; d tj = 0, when
«#;)■
With a u and c tj computed, the problem can then be solved by methods
in Chapter 3. We solve a problem of similar nature in Art. 4.8.
4.8 Formulation of Problem by InfiniteSeries Expansions
of Energy Expressions — RayleighRitz Method Reexamined
(A) SOLUTION BY INFINITE SERIES
In Art. 4.1 we mentioned the alternate method of specifying the con
figuration of a vibrating elastic body by infinite series. We now show that
this method leads directly to an approximate solution of the free vibration
problem, which resembles the RayleighRitz method in detail.
Consider first the simple case of longitudinal vibrations of a slender
bar of length L. The motion of the system is described by the displace
ment function u(x, t), which is the unknown to be determined. At any
T m = \mco 2 u 2 dx
Jo
= i^ViM?
a i} = m
Jo
7TX~
1 _ cos(2 /l)_
1 — cos (2/ —
7TX
2mLt
77 >
tt (1)' (iy
2 2/  1 2/  1
+<)
288 THEORY OF MECHANICAL VIBRATION (4.8)
instant t the shape (or the configuration) of the bar is described by a
function of x. As a function of x for a fixed t, u is known to be (i) con
tinuous for < x < L, since the bar would otherwise be broken, and (ii)
E(dujdx) = a x exists and is continuous in < x < L, except possibly
at a few points along the bar where concentrated external forces are
acting on it.
It is known that all functions having the regular behavior described above
can be expanded into an infinite series formed by a set of "suitably
chosen" functions of x: l5 </> 2 , ...,</>.,... .
u = ^ + q 2 cf> 2 , • • • , q^ • • •
so that the series converges uniformly in < x < L and is termwise
differentiate, except possibly at a few points. The same set of functions <z>
can be used for different values of t. As u varies with t, the ^'s vary also.
Therefore,
00
i = l
The unknown of the problem now becomes a set of functions of t: q v
q 2 , . . . , q it . . . which now becomes the generalized coordinates of the
system, since by specifying their values one specifies uniquely the con
figuration of the system.
To study what constitutes a suitable set of </>'s, we must consider among
other things the boundary conditions of the bar. Let us assume that the
bar is held fixed at the left end where x = 0. Hence
W (0, t) = ^(r)^i(O) + q 2 (t)U0) • • • =
for all /. If the ^'s, the generalized coordinates, are supposed to be
linearly independent, it becomes necessary that
</>,(()) = (100)
for all /'.
We know that a continuous function in < x < L can be expanded into
a Fourier series with the fundamental interval — 2L < x < 2L. We choose
a fundamental interval of expansion longer than the length of the bar L
so that we will have more selections of the functions to be used. 22 To
satisfy (100), only the sine functions are retained.
sin — — n = 1, 2, 3, • • •
22 Also, by making x = and x = L inside the fundamental interval of Fourier
series expansion, it is easier to make the series converge to the function // at these two
points.
(4.8) VIBRATION OF ELASTIC BODIES 289
Furthermore, we can delete from this set either those with odd n or
those with even n, depending upon the boundary condition at the right end.
If the right end is also held fixed, that is,
u(L) =
to satisfy this imposed boundary condition, we must choose n as even.
If the right end is allowed to move (not necessarily free), we must choose n
as odd, so that the sine series can converge at x = L to a value other than
zero.
Let us suppose that at the right end a concentrated mass M is attached.
The boundary condition there is
, du d 2 u
k— = M— at x = L (101)
ox ot A
in which k = EA. Evidently this condition is not satisfied by
. YlTTX
sin — n = 1, 3, 5, • • •
2L
But this is not important for reasons to be seen later.
The kinetic energy rand potential energy Kfor this system are given by
1 [ L (du\ 2 , 1 (du\
If we represent u(x, t) by
T
) [.. \ 'At ! ' '") \ 'At I
x = L
i r L /du\ 2
V= I k\ — \ dx
(102)
w= ^ Wsin (?Lzi^ (1 o3)
?=i IL
we can obtain dujdx by termwise differentiation,
du " (2il)7T (2il)7TX ritXA ^
~=I qlt) cos —L (104)
ox 1=1 2L 2L
which is valid for all < x < L. We can also obtain dujdt by termwise
differentiation,
du " e (2i \)ttx
= StfXOsin  lr (105)
ot i=\ 2L
290 THEORY OF MECHANICAL VIBRATION (4.8)
assuming that the resulting series is uniformly convergent with respect to t.
By substituting (104) and (105) into (102), we obtain
1 CO CO
Z i = 1 3 = 1
(106)
i oo co '
Z i = l j = l
where
f L , , • (2/  1)ttx . (2/  1)ttz , . .
a tj = m(«) sin — sin dx + {\)' J M
Jo 2L 2L
c „ = JV)^(2/l)(2y l)cos ( ?^ 7 l^cos^ z ^*.
Now the conservation of energy requires that
^(r+ k) = o
CO oo
2 2 (««& + ctfMt = o
i = l j=l
Since g's are supposed to be independent,
CO
2 arf, + c ijqj = i = 1, 2, 3, • • ■ (107)
i=i
Thus we have an infinite set of linear differential equations of an infinite
number of unknowns.
Whether or not such infinite formulation is always mathematically
legitimate is too involved a question to be discussed here. For a given
problem this question may be answered when the solution is obtained and
each step is retraced to examine its legitimacy. Even that task is difficult.
Fortunately, as subsequently demonstrated, the reasoning used here leads
in practice to the same result as the RayleighRitz method, of which the
mathematical foundation is more secure.
We note that each of the function <j> t in (99) satisfies the imposed
boundary conditions, hence can be used as a generating function in the
RayleighRitz method. If we terminate (99) at the /2th term, the energy
expressions (106) will be that of an ^degreefreedom system. The fre
quency equation obtained from (107) with i,j= 1,2,..., //, is exactly that
for minimizing (96).
(4.8)
VIBRATION OF ELASTIC BODIES
291
As an example, let us take the simple case in which m and k are con
stants in (102). The resulting problem was solved exactly in Art. 4.3(a);
therefore, we can evaluate the accuracy of the method. Let
n = 4
and
M
\mL
. TTX ZlTX 5tTX IttX
u(x, t) = ft sin 2l + q2 Sin ~2L +%Sm ~2L + q * Sm ~T
By putting this into (106), we obtain
T = \mL[qf + q 2 + q 2 + q 2 + (q x  q 2 + q 3  q^f]
= \mL(2q^ + 2q 2 + 2q 2 + 2q 2  2q x q 2 + 2q 1 q 3
 2q x q±  2q 2 q 3 + 2q 2 q^  2q 3 q 4 )
}*E« +
, f i + 25 ft « + 49? 4 2 )
Hence
A =
2 1
1 2
1 1
1 1
1
1
2
mL
c =
1
0"
9
kjr
25
%L
49
C^A
2.0000
0.1111
0.0400
0.0204
■1.0000 1.0000 1.0000"
0.2222 0.1111 0.1111
■0.0400 0.0800 0.0400
0.0204 0.0204 0.0408
4mL 2
By the matrix iteration method described in Chapter 3, we obtain
o
and
\M6V(klm)L = \M6\/(EI P )L
TTX 3lTX 5TTX ItTX
r, = 92.8 sin 5.68 sin h 1.99 sin sin
1 2L 2L 2L 2L
The exact solution of (24) for jli = \ is
o h = l.0169V(Elp)L
and
r x = Bs'm 1.0769
in which B is an arbitrary multiplication constant. The comparison
between i\ and T\ is shown in Fig. 93.
292
THEORY OF MECHANICAL VIBRATION
(4.8)
Note that the approximate function /\ does not satisfy the boundary
condition (101). This is reasonable, since that boundary condition
describes an equilibrium relationship that supplements the differential
equation of motion. In other words, both the differential equation (5) and
the dynamical boundary condition (101) originate from Newton's law.
In the infiniteseries formulation they are replaced by (107). An approxi
mate solution of (107) is not expected to satisfy (101) any more than it is
expected to satisfy the differential equation of motion. 23 However, by
Figure 93
taking a sufficient number of terms in (103), the two sides of (101) can be
made arbitrarily close to each other at x = L — e, where e is arbitrarily
small.
(b) effects of constraints on natural frequencies
A very important concept in vibration analysis is that the introduction
of additional constraints into a system will raise all the natural frequencies
of the system (or at least will not lower them). For instance, in the lateral
vibration of bars all the natural frequencies will be raised if a free end is
made into a hinged end, and they will be raised again if the hinged end
becomes a builtin end.
To prove this statement in a general way, we utilize the maximum
minimum characterization of eigenvalues discussed in Art. 4.6(c). Before
we proceed, let us point out that an imposed boundary condition, being
geometrical in nature, constitutes a constraint, whereas a natural boundary
23 In this particular example the approximate solution does satisfy the differential
equation of motion. This is an exception rather than a rule.
(4.8) VIBRATION OF ELASTIC BODIES 293
condition is not considered as a constraint, since it originates from con
sideration of forces. A constraint may also be any interdependence
relation imposed upon the coordinates of a system; for instance, such
imposition as
w(0, t) = 3w(L/2, /) = u(L, t)
or in (99)
<7i = % = <?5 = ' ' • =
Let S be a vibrating system and S be another vibrating system, which
can be obtained by imposing additional constraints on S. The two
systems therefore share the same differential operators L and M, and their
energy expressions derived from an (essentially) admissible function will
also look the same. Hence Rayleigh's quotients for the two systems are
computed in the same way. We note also that since any constraint of
system S is also a constraint of S, but not vice versa, any admissible or
essentially admissible function of § is also admissible for 5, but not vice
versa.
Let r l9 r 2 , . . . , r fc _i be the first k — 1 eigenfunctions of S. Let C k be the
class of functions that is admissible to S and orthogonal to t\, r 2 , . . . , r k _ l9
with respect to M. Let C k be the class of functions that is admissible to S
and orthogonal to r ls r 2 , . . . , r k _ v It is clear that every function in C k
must also belong to C k , but not vice versa. According to the minimum
characterization of eigenvalues, the function in C k having the least
Rayleigh's quotient is the kth eigenfunction of S, namely r k , and
fi(W) = <■>*
Since C k contains all the functions in C k , the minimum value of Rayleigh's
quotient formed by a function u in C k is no less than Q((r k )).
lOmirtn > 2((0) = <
In the meantime, if co k 2 is the kth eigenvalue of S, according to Art.
4.6(c),
*V > [Q((u))] mm
because C k is formed by removing from functions admissible to Sail those
which do not satisfy
r.M(w) da = ] = 1, 2, • • • , (k  1)
and, insofar as S is concerned, r l9 r 2 , . . . , r k _ 1 is a set of arbitrary functions.
Hence
c5 7c 2 >fo fc 2 (108)
294 THEORY OF MECHANICAL VIBRATION (4.8)
The natural frequencies obtained by the RayleighRitz method can also be
considered as the true natural frequencies of a hypothetical system, which
is obtained by imposing additional constraints to the actual system.
By terminating the infinite series (99) at the nth term, we introduce the
constraints
q n+1 (t) = q n+2 (t) • • • =
We note also that in proving (108) all we needed was the fact that the class
of admissible functions of S is a smaller class than that for S. It is therefore
clear that the n natural frequencies obtained by the RayleighRitz method,
interpreted in the light of either Art. 4.7 or Art. 4.8, must be equal to or
greater than the first n frequencies of the actual system. Hence
and by retaining more terms in (99) or by adding more functions to the
generating set for (82) the answers tend to be improved. In other words.
the RayleighRitz method approximates the natural frequencies from above
and can always be used to establish their upper bounds.
(C) ADDITIONAL REMARKS ABOUT ENERGY METHOD
(i) From the simple example worked out in (a) it is seen that the energy
method can give usable engineering answers for the lowest natural fre
quency with a relatively small, wellchosen generating set. The eigen
function obtained from the approximate method is, however, not quite so
reliable, especially at places near a boundary point at which a natural
boundary condition is not satisfied by the functions in the generating set.
A look at Fig. 93 will reveal this fact.
(ii) If n lowest natural frequencies are to be found, a generating set
larger than n functions is needed. How much larger the generating set has
to be depends on the nature of the functions chosen. If they have close
analytical resemblance to the eigenfunctions of the problem, only a few
more than n functions are needed. The existence of such resemblance can
be assured if the diagonal elements of both A and C for the approximating
system are much larger than their offdiagonal elements. Although this
condition is not mathematically necessary for the generating set to
generate functions closely approximating eigenfunctions. in practice we
always aim to achieve this condition.
(iii) It must be pointed out that in practical problems we are seldom
interested in knowing more than the first few natural modes of the system.
Furthermore, although the energy method combined with a digital
computer can determine accurately the higher eigenvalues and their
associated eigenfunctions, they may not describe the actual modes of
(4.8) VIBRATION OF ELASTIC BODIES 295
vibration. This is because the mathematical formulation often loses its
physical validity for these modes. For instance, the first few modes of the
lateral vibration of a round bar having a lengthtodiameter ratio of 10 can
be described accurately by the equations or energy expressions obtained
from the elementary theory of beams. But these same equations and
energy expressions are grossly in error when applied to, say, the tenth or
higher modes.
(iv) The labor required as well as the accuracy obtainable in solving
problems approximately by the energy method depends upon the choice of
generating functions <f> x , </> 2 , . . . , <j> n . A desirable choice is one which
facilitates the evaluation of the integral in (83) and yields a pair of matrices
A and C, whose diagonal terms are much larger than the offdiagonal
terms. In other words, we aim to make the </>'s and the true eigenfunctions
as nearly alike as possible, without excessive computational labor.
There are no hard and fast rules to be followed in achieving these
results. The following observations, however, may serve as a rough guide.
(i) Polynomials are often conveniently chosen as generating functions
because they can be easily differentiated, integrated, and made to satisfy
the boundary conditions. For instance, in analyzing a nonuniform beam
with a builtin end at x = and a free end at x = L the generating set can
be simply
/yiLt /yd /y**± ■ • • /y»'t
which satisfy the imposed boundary condition at x = but not the natural
boundary condition at x = L. Or we may use as the generating set
fa = x 4  4Lx s + 6L 2 x 2
fa = x 5  \0L 2 x* + 20L 3 * 2
fa = x 6  20L 3 x* + 45ZAr 2
fa = x i  35L 4 x 3 + 84LV
which satisfy all the boundary conditions.
Tt is to be expected that for a given accuracy we need fewer functions
of the second set than will be needed if the first set is used. The matrix
problem resulting from the second generating set will be smaller but it will
take longer to compute a u and c u defined by (83).
(ii) It is always desirable to take advantage of any known orthogonal
properties relative to the differential operators M and L. For instance, if
M is merely a simple multiplication constant, then the generating set
can be simple sine and cosine functions satisfying the imposed boundary
296 THEORY OF MECHANICAL VIBRATION (4.9)
conditions. In that way, a {j = 0, when i^=j; or, if M = x, we may
choose Bessel functions of the appropriate order satisfying the imposed
boundary conditions.
(iii) If by changing the natural boundary conditions of a problem the
eigenfunctions are known, these eigenfunctions can be advantageously
chosen to be the generating functions of the original problem. For example,
the eigenfunctions (36) of the cantilever beam shown in Fig. 88 are good
generating functions for the system in Fig. 92, especially when the right
end spring supports are soft. Similarly, if by a slight change of the
differential equation the eigenfunctions of the problem are known, they
can be used as the generating functions of the original problem.
4.9 Forced Vibration of Elastic Bodies
(a) concentrated force applied at a fixed point
In Chapter 3 we discussed a method of solving forcedvibration problems
by a transformation of coordinates that results in a set of differential
equations, each of which contains only a single unknown, namely, one of
the principal coordinates. The same approach has proved to be a con
venient one for the problem at hand.
The essence of the discussions in Arts. 4.7 and 4.8 is that the distribution
of elastic force and that of the inertia force in a vibrating elastic body can
be characterized either by two differential operators L and M. together
with a set of boundary conditions, or by two energy expressions Fand T.
Furthermore, if u is the actual displacement function during vibration,
the energy expressions and the differential operators are related by
= *J>
M(w) da
(109)
V = \ I ul(u) da
J 2
where u stands for dujdt.
In free vibrations the principle of conservation of energy
l(T+ V) =
at
leads to the equations of motion. Similarly, we may derive the equations
of motion for forced vibrations from
— (T + V) = rate of doins work by external forces
dt
(4.9) VIBRATION OF ELASTIC BODIES 297
For a given elastic system the energy expressions V and T, as well as the
operators L and M, are the same regardless of whether the vibration is
forced or free.
Let u be the displacement function describing the forced vibration of an
elastic body. Let the eigenfunctions for the free vibration of this elastic
body be r lt r 2 , . . . , r,, ... . The function u can be expanded into an
infinite series of r's.
<x, t) = fp#Mz) (110)
i = \
We assume here that there is only one space variable, x. However, the
generalization to cases with more than one space variable is immediate.
As the eigenfunctions in free vibration, the r's have the property
f r,M(r 3 ) dx =
Jo
i^j (HI)
L
/o
and
L(r,) = a>/M(r,) (112)
I
Now let us call for simplicity
rW da = m u (113)
L
By substituting (110) into (109) and utilizing (111), (112), and (113), we
have
oo oo
Hence
d °°
 (T + V) = 2 m^ + w,.^
a / { = i
Let a force/(?) in the direction of u be applied at a fixed point in the body
at x = £. The rate of doing work by this force is
/(oco=/wIm«)
Therefore, the energy equation leads to
i [m«(A + o>i 2 Pi)  foymipi = o (H4)
298
THEORY OF MECHANICAL VIBRATION
(4.9)
Since p/s are not dynamically related to p t or p i9 being dependent upon
initial conditions which are quite arbitrary, the vanishing of (1 13) demands
that
m u (pi + cofc) =f(t)r i (i) i = 1 , 2, 3, • • • (115)
There is an infinite number of equations in (115), but each contains only
one unknown, which can be found separately by the methods discussed in
Chapter 1 . The initial conditions for p i are derived from those for u.
oo
i = i
\Ot/t = t i=l
After the/?'s are obtained, they are put into (1 10) to get the solution for u.
Figure 94
The solution is in the form of an infinite series. This series must be
convergent and termwise differentiable in order that the operations leading
to (114) legitimate. Such is usually the case.
To illustrate the procedure, let us take the case of a simply supported
beam, having uniform cross sections and subjected to a concentrated force
f(t) at x = £, as shown in Fig. 94. According to the results on p. 257
the eigenfunctions for this case are the sine functions
17TX
r, = sin
and the eigenvalues are given by
and
CO
m
t _EW_EI/*i\
1 " P A " pA\l)
« = f "
2 f7TX A L
sin' 2 — — ax — —
2 2
for all i
(4.9) VIBRATION OF ELASTIC BODIES 299
By substituting these into (115), we have
EI /tt\ 4 a 2 . fai .,
P A
(b) DISTRIBUTED EXTERNAL FORCE, MOVING EXTERNAL FORCE,
AND GENERALIZED EXTERNAL FORCE
The results given in (a) can now be generalized in the following three
ways:
(i) If the external force is distributed over the elastic body with the
distribution described by
/=/(*,o
(115) is modified into
mJPi + cofc) = f /({, t)rm dS (116)
Jo
by the principle of superposition.
#
W
Figure 95
(ii) If a concentrated external force changes its point of application,
then £ = £(/), and (115) is changed into
m u (p t + o>* Pi ) =f(t)rlHt)]
This is true because there is nothing in the derivation of (115) that implies
I is fixed. This generalization is of practical interest in studying vibrations
of a structure under a moving load. For example, in Fig. 95 a weight W
moves across the simply supported horizontal beam with a uniform
velocity v, and the force exerted on the beam is of magnitude
Wa u ,=f(t) (117)
g
in which a w is the vertical acceleration of W. If a w <^:g, (115) is modified
into
mdpi + w«Vi) =  Wrgvt + f )]
300 THEORY OF MECHANICAL VIBRATION (4.9)
in which £ is the location of W at t = t . Otherwise
(d 2 u „ d 2 u av\
x=
i = i
By substituting this expression for a w in (117) and (115) and replacing £
by (*;/ f  ), we obtain a set of linear differential equations for /?'s in
which the coefficients vary with time. Furthermore, all the unknowns
appear in all the equations. Such equations are not readily solvable. They
are mentioned here merely to illustrate the fact than when additional
inertia force due to Sexists the original set of eigenfunctions is no longer
that of the new system, and since this additional inertia force changes its
location the system is no longer timeinvariant.
(iii) Generalized forces. In the preceding examples we assume that
work is done by external forces applied in the direction of the displace
ment w, and for a single concentrated force at x = £ the rate of doing work
is
d ,„ . _ „_Jdu\
A (r + K)=/(0L
But there are other ways in which external forces may do work. Taking
the lateral vibrations of bars as an example, we know that work can be
performed by a couple or moment in the plane of bending of the bar. In
that case
d ._ __ _..Jl d 2 u
It
(r+K)«(0(^) S^Or/WO
Thus (115) is modified into
mdpi + co 2 Pi ) = M(t)r^)
Work can also be performed by a pair of equal and opposite axial
forces acting on two ends of the bar, as shown in Fig. 96. Let the force be
P(0; then
1{T+ V) = P(t)X(t)
at
where X(t) is the displacement relative between the two ends. To compute
X{t), we observe that two points on the neutral axis a distance dx apart
(4.9)
VIBRATION OF ELASTIC BODIES
301
are separated in the axial direction during vibration by a distance dx cos f),
where Q is the slope of the deflected bar. 24 Hence
d C L d C L 6 2
X(i) =  dx (1  cos 0) = — dx —
It
Jo 2\dxJ
dx
P(t)
dX = dx(l cos 0)
Udx J
Figure 96
By utilizing (1 10), we have
*c) = ii piPi f^'wc*) *
i = 1 i = 1 Jo
This modifies (1 15) to read
™u(pi + <o*Pi) = P(t) 2 Pi r/(xy/(x) dx
i=i Jo
The introduction of axial forces is similar to a change in the elastic
property of the system. As a result, the r's are no longer the eigenfunctions
and the/?'s no longer appear separately in the equations. In the particular
case of a uniform bar with two hinged ends
. 17TX
r£x) = stn —
, in 17TX
r„ (x) = — cos — .
1 W L L
it so happens that
r
r/(xy/(x)dx = i^j
I
o 2L
24 The effects of the direct compressive stress produced by P(t) are neglected.
302 THEORY OF MECHANICAL VIBRATION (4.9)
Here again we have a set of equations with variables separated:
*2 2
or
m u pi + 1%^  ^ ?(*)) Pi =
Since for this particular case
we have
H = M sin 2 l ^dx = ?—
Jo L 2
/O^L i 2 7T 2 (i 2 7T 2 EI
If P is a constant, the foregoing equation can be considered as the standard
form for systems in free vibration.
Pi + &*Pi = °
where
i 2 7T 2 i 2 7T 2 EI \
P AL 2 \ L 2 )
ojs =
In other words the existence of an axial force merely modifies the elastic
force of the system; hence it changes the natural frequencies. It is inter
esting to note that if P is equal to Euler's critical load for column buckling
7T 2 EI
P = [?
then e^ = 0, and the system is in a neutral equilibrium. If P is less than
Euler's critical load, all the natural frequencies will be real and positive,
and the system is stable. If P is greater than Euler's critical load, one or
more of the natural frequencies will be imaginary, and the system is
capable of a spontaneous motion in which the displacements will grow
without limit. The system is therefore unstable. The fact that a column
subjected to axial forces greater than Euler's buckling load is an unstable
column is thus explained by theory of vibration, whereas it cannot be
explained by elementary column theory. We can therefore state that the
criterion for the stability of an elastic system is the absence of negative
eigenvalues or imaginary frequencies in the free vibration problem of the
system.
(4.9) VIBRATION OF ELASTIC BODIES 303
(C) INDICIAL RESPONSE, IMPULSE RESPONSE, AND IMPACT —
GENERAL SOLUTION OF FORCED VIBRATION PROBLEMS
The indicial response of an elastic system can be defined as the vibration
caused by a concentrated unit step force applied at a fixed point in the
system and at a time when the system is at rest.
Let this indicial response be denoted by the function £/(£, x, t) for a
onedimensional elastic system. If its eigenfunctionseries expansion is
U&x,t) = fplt)rlx)
i = l
then p i is the solution to
m ii {p i + M ?p t ) = S(t)r^) i = 1, 2, 3, • • •
under the initial conditions that
P M = o /),(0) = o
According to (75) of Chapter 1,
Hence
PAD = ^ 2 (i  cos «v)
m u cof
i=i rn ti (o t 2
The impulse response, which describes the free vibration caused by an
impact, is obtained by differentiating U with respect to t.
Thus, if the system is initially at rest and is subsequently set into motion by
a force /(0 applied at x = £, the solution is
00 r (E\r (x\ C f
u(x, 0=1  — /(t) sin co t {t  r) dr (118)
i = i m ii p i Jo
And, if we are dealing with distributed force of distribution g(£, t), the
solution is
u(x, t)
Z sin co t (t — r) dr d$ (119)
ii Jo Jo rnuoii
This is the general solution of forced vibration of a onedimensional
elastic body which is initially at rest.
304 THEORY OF MECHANICAL VIBRATION (4.10)
4.10 Vibration of an Infinite or SemiInfinite Elastic Body —
Wave Phenomenon
(a) continuous frequency spectrum
The systems we have dealt with so far are finite in extent. A system of
this kind was found to possess a set of distinct natural modes, and its
natural frequencies form a discrete set of values. As a result, any free
vibration it may have is representable by an infinite series
oc
m = 2Q cos (cos — <x l )r i (120)
The total energy possessed by the system in a given vibratory motion is
T+ V= U [uM(u) + ul(u)]da
J 2
= i!m ii C i W (121)
i = l
where m u is defined in (113). The total energy expression (121) indicates
that a unique and definite amount of energy is associated with each natural
frequency or that between different natural modes there is no energy
transfer. Because the natural frequencies are discrete the free vibrations of
a finite body are said to have discrete frequency spectra.
To examine the changes when an elastic body becomes infinite or semi
infinite in extent, let us reflect that for a finite body the eigenvalues or the
natural frequencies are determined by the conditions at the boundary.
When a system becomes infinite or semiinfinite, all or part of its boundary
is located at infinity, where the prevailing conditions are hardly expected
to have an effect on the motion of the interior mass elements of the
system. Take, for example, the longitudinal vibration of a uniform bar
with a fixed end at x = 0. The eigenfunction is
r(x) = sin fix
The allowable values for (3, or the eigenvalues, are determined by a linear
homogeneous boundary condition at x = L.
ar + br' = a sin /3L + bfi cos pL = (122)
This becomes the characteristic equation of the problem. Now. as L
becomes infinitely large, this equation can be satisfied by any arbitrary
real value 25 of fi. Looking at the problem in another way, we see that the
25 To state it correctly, we should say that given an arbitrarily small e > 0. with a
sufficiently large L we can find for every real positive fi a ft such that /3 satisfies (122)
and \p  j5 < e.
(4.10) VIBRATION OF ELASTIC BODIES 305
characteristic equation for a onedimensional continuum is satisfied by
PL = (one of a set of dimensionless numbers). The difference between two
consecutive numbers is always finite.
Pi+iL ~ PiL < N < oo for all i
Hence
lim (/?, +1  ft) = for all i
L— > oo
All this means that when an elastic body is infinite in extent its frequency
spectrum becomes continuous.
(b) a phenomenon of wave propagation
The concept of the natural mode of vibration loses its physical signi
ficance in dealing with an infinite or semiinfinite body, not only because
the natural frequencies are no longer distinct, but also because the vibra
tion represented by an eigenfunction cannot be excited without an infinite
amount of energy input. For instance, a longitudinal vibration represented
by
u(x, t) = C sin fix cos (cot — a)
p = coVe/p c>°
is mathematically possible for a uniform bar of infinite length. But such
vibration is of little physical interest, since its energy content is infinite as
1 f °°
(J + V) =  co 2 C 2 sin 2 Bx dx = oo
2Jo
A more realistic physical problem follows. An infinite elastic medium is
initially at quiescent state. A finite disturbance is introduced at certain
parts of the medium. What is the resulting motion of the medium?
Consider again the longitudinal vibration of a uniform bar, which is
governed by the onedimensional wave equation :
d 2 u 3 2 u
8?= C *W (123)
where
c = + Ve[p
The general solution to this secondorder, partial differential equation is
u = (f)(x  ct) + ip(x + ct) (124)
306 THEORY OF MECHANICAL VIBRATION (4.10)
in which <j> and tp are two arbitrary, twicedifferentiable functions. The
way x and t enter into the arguments of <f> and ip indicates that a propaga
tion phenomenon is involved. Obviously,
c/)(x 1 — cti) = <f>(x 2  ct^)
if
x 2  x 1 = c(t 2  tj)
Stated in words, these equations say that whatever exists at x Y will exist at
x 2 at an instant (x 2 — x^/c later. The phenomenon represented by <f>
therefore propagates to the right. Similarly,
ytei + ct Y ) = yj(x 2 + ct 2 )
if
x 2 x 1 = c(t 2  tj)
which says that whatever exists at x 1 existed at x 2 at an instant (x 2 — x^)jc
earlier. The phenomenon represented by ip therefore propagates to the
left. In either case the speed of propagation is the constant c.
Now let us suppose we have a very long bar extending from x = a <
to x = b > 0. A small external disturbance is introduced into the bar
near x = at t = 0. Let this disturbance be in the form of an initial
displacement, which is restricted to a small interval containing the point
x = 0. The initial conditions are thus
u (x, 0) = U(x)
(125)
KL
As shown in Fig. 97, the initial displacement function L\x) is zero,
outside of a small interval. It can be readily verified that both the differ
ential equation (123) and the initial conditions (125) will be satisfied if we
let
in (124), or
u(x, t) = ±U(x ct) + \U(x f ct) (126)
Hence the disturbance introduces splits into two equal waves and pro
pagates in two opposite directions with a constant speed c. If the bar
extends to infinity at both ends, then (126) is valid for all r's. If, on the
other hand, the bar terminates at a finite point or points, the solution
(126) is valid for a time interval < t < t l9 in which r x is time required for
the disturbance to reach one or both ends of the bar. This is so, regardless
(4.10)
VIBRATION OF ELASTIC BODIES
307
Free end
U_J
\^2e^\
2e
z^
U2e
l^2e^ U — 2e — J
Fixed end
/\
Figure 97
Figure 98
308 THEORY OF MECHANICAL VIBRATION (4.10)
of the end conditions, because, according to (126), until the disturbance
reaches an end the mass element at that end has zero displacement and
zero velocity, a combination that satisfies any homogeneous boundary
condition.
If the bar has an end, the disturbance wave will be reflected as it reaches
that end. In a certain subsequent time period the reflected wave and the
incident wave coexist in the bar. Afterward, only the reflected wave
remains. The manner in which the reflection takes place depends on the
end condition. At a fixed end
"incident ' "reflected "
and at a free end
incident \ OX] reflected
These conditions are graphically illustrated in Fig. 98. Because neither
a fixed end nor a free end can absorb energy, the reflected wave has the
same shape and energy as the incident wave.
(C) NATURAL MODES AND STANDING WAVES
Consider now a bar having fixed ends at x = —L/2 and x = L/2.
An initial disturbance is introduced into the bar. This disturbance splits
and propagates toward the ends with speed c. It is evident that each of
the two parts will return to its original location, shape, and direction of
propagation after being reflected once at each end. After an elapsed time
of 2Ljc, the two parts reconstitute the original disturbance. This elapsed
time thus becomes the fundamental period of the natural vibration
t = — = ilV^Je
c
If the bar has a fixed end and a free end, each of the split disturbances will
have to be reflected twice by each end to return to its original condition.
The period is then
T = — = ALVpjE
c
Take a special case in which the initial disturbance for a bar having two
fixed ends is given by
1TX
U{x) = C sin —
(4.10)
VIBRATION OF ELASTIC BODIES
309
Referring to Fig. 99, we see after the wave splits that the reflection from
one part complements the incident wave of the other part to constitute a
shifted half sine wave. Hence for this special case (126) is always valid.
Therefore,
7T 77"
u(x, t) = C[sin  (x — ct) + sin  (x + ct)]
J/ Li
TTX TTCt
= C sin — cos —
Li Ld
t =
t = t
ct,^>
Figure 99
Since the shape of the displacement curve is now always a half sine wave,
we call it a standing wave, and it represents the first natural mode of
vibration. A standing wave can also be formed in this case by
mrx
U(x) = C sin
in which n is any integer.
(D) LATERAL VIBRATION OF AN INFINITE BAR
We now illustrate another approach to the analysis of vibrations of an
infinite body. A mathematically rigorous development of the subject to be
discussed must utilize the theory of Fourier transformation. Here we have
310 THEORY OF MECHANICAL VIBRATION (4.10)
to be satisfied with a version that is mathematically less rigorous but
physically more informative.
Let us begin with a finite bar extending from x = — L/2 to x = 12.
According to (37), the free vibration of this bar can be described by
co
v(x, t) = 2 C, cos (a),/  a^rjix) (37)
Since for each i there is a ft, we may consider C, co, a, and r as functions
of a variable ft Let us rewrite (37) as
v(x, t) = 2 rf cos (co,/  a,K(*) A/?, (127)
*i Aft
where Aft = ft — ft_ r We noted in (a) that as L + co, ft + 0, Aft > 0.
and Q+0. If the limit
c.
lim —4 = lim £, < co
L^ooAft
exists, the righthand side of (127) becomes an improper integral as L > x
or Aft^0.
*;(*, 0= £(/?) cos {cot  a>(ft a;) dfi (128)
Jo
where
co = pWEJjpA (129)
This solution is arrived at strictly in a formal way. We have no knowl
edge of the appearance of E(ft\ oc(ft, and r(ft x), nor do we know whether
or not the improper integral converges.
To find out how (128) can be made to fit the problem, we observe first
that if the partial differential equation (28) is to be satisfied by (128)
r(ft x) must satisfy (30). Hence from (32)
r(ft x) = A sin fix + B cos fix + C sinh ftc + Z) cosh ftr
in which A, B, C, and D may be functions of ft but not of .r. Since (128)
must converge for all x and since hyperbolic functions grow without limit,
the coefficients C and D must vanish. Thus
r(ft x) = A sin fix + B cos ftz
= F(ft cos (/5k  6)
By substituting this into (128) and combining E and .Finto a new function
G, we have
v (x, 0= G(/?) cos (ft)/  a) cos (ftr  d) dfi (130)
(4.10) VIBRATION OF ELASTIC BODIES 311
Now let us take a simple initial condition
v (x, 0) = V(x)
Furthermore, let us assume that Vis an even function, V(x) = V(—x). It
is not difficult to see that these conditions imply
<x(/?) = and d(p) =
K(s) = I °°G(/5) cos fix dp (131)
Jo
This is the formula for a Fourier cosine transform, 26 and the inverse is
given by
2f co
G(P) =  V(x) cos (fix) dx
ttJo
The solution is then
v(
x,t)= G(P) cos cot cos fix dp (132)
Jo
where co is given by (129). By an identical procedure, we can show that if
the initial displacement is an odd function of x
v (x,0) = V(x) = V(x)
d = tt/2
1
v(x, t) = G(P) cos cot sin fix d/1 (133)
then
and
where
G(f$) =  I V(x) sin ^x dx
77 Jo
Since any arbitrary initial displacement function can be decomposed into
an even function and an odd function by
V(x) = i[V(x) + V(x)] + \[V(x)  V(x)]
by the principle of superposition, we can obtain the solution for any
initial displacement V(x). The result is
77 Jo J
v(x, t) =  V(i) cos 6(£  x) cos cot d£ dp (134)
26 See Sneddon, Fourier Transforms, McGrawHill, New York, 1951, p. 1!
312 THEORY OF MECHANICAL VIBRATION (4.11)
4.11 Method of Finite Difference
A very straightforward way of solving an eigenvalue problem numerically
is by the method of finite difference. The principle is simply that differen
tial equations and integral equations can be approximated by algebraic
equations. The technique, however, has many refinements and ramifi
cations. In this article we shall illustrate the basic principle by simple
examples. 27
(a) finite difference approximation for a differential equation
Consider the eigenvalue problem in the lateral vibration of uniform
bars. Formulation by differential equation leads to
S^0 (135)
Let (n + 1) equally spaced points be chosen along the bar. These are
called the pivot points: x = 0, x x = h, x 2 = 2/z, . . . , x n = nh = L. Let
the corresponding values and the derivatives for r be designated by
r = r(0), r x = r(h), r 2 = r(2h), ■•,/•„ = r(L)
r ' = r'(0), r{ = r\h\ • • • , r n ' = r\L)
For the purpose of deriving formulas we must also introduce the
following intermediate pivot points:
_h _3/z
Xl A ~ 2 Xiy * ~ ~2 etC '
The derivatives of r of various orders can then be approximated by the
following difference formulas
hh" = h(r' i+1A  r\_ v ) = {r i+1  r t  r< + r^J
= r i+1  2r L + r t ._!
= r i+H  2r i+H + r^ H  (r i+1A  2r t _^ + r t _ SA )
= r i+H  3r ( +y 2 + 3r { _ 1A  r i _ S/i (136)
It is not difficult to show that the formulas resemble that of binomial
expansion for
a%a A  a~ 1A ) j
27 See also the example at the end of Art. 4.2(a).
(4.11) VIBRATION OF ELASTIC BODIES 313
in which the exponents of a become the subscripts for r and the coeffi
cients in the expansions are the same. At any rate, the approximation for
the fourthorder derivative needed for the present problem is
d*r „„ 1
^a = r i = ^i fo+2 ~ 4r i+i + 6r i ~ 4r ii + r i2)
/= 1,2, ••,(/! 1)
We can thus approximate (135) by
r i+2 ~ 4^+i + 6r t  4r,_ ± + r,_ 2 = (hpfr, / = 1, 2, •••,(« 1)
(137)
There is one such equation for each interior pivot point.
Besides the interior pivot points, there are two boundary points.
Moreover, to write the difference equation (137) for an interior pivot
point near the boundary, we shall also need two imaginary pivot points
lying outside the interval to L. The values of r at these four points must
either be known or be expressed in terms of those for the interior points
by the use of the boundary conditions.
At a hinged end,
or r , =
r(0)
=
thus r =
r"(0) = thus
fi
 2r + r x =
At a builtin end
r(0)
=
thus r =
r'(0)
=
thus r x — r_
At a free end the situation is slightly complicated. There we try to
write (137) differently for the nearby interior pivot points so that they
will not include r_ x and/or r . This is done by the following scheme.
The fourth derivative of the first pivot point may be written
Since r
— (r 3  2r 2 + r x  2r 2 + Ar x  2r )
(138)
j£ (r s  4r 2 + 5/J  2r )
314 THEORY OF MECHANICAL VIBRATION (4.11)
In the meantime, the same derivative can also be written
* / in Wx *■
Th (2 ~ t{i) ~ih
/ /// lll\ y III III r\
2  r o ) = —,. r 2 smce r =
where we may approximate r% by
r" — Ur'" 4 r'" \
r 2 — 2V2.b + r l.5>>
Utilizing (136), we obtain
(r 4  2r 3 + 2r x  r ) (139)
4/z 4
Elimination of r between (138) and (139) yields
r'i = ^ (2r,  5r 3 + 4r 2  rj = /?% (140)
This equation now replaces the first equation in (137). Also by equating
(138) and (139), we can express r in terms of r ls r 2 , r 3 , and r 4 . The result
ing expression can be used to eliminate r in the second equation of (137).
Example: To find the natural frequency for the first mode of lateral
vibration of a uniform bar with hinged ends.
Let the spacing of pivot points be
*§
Because of the symmetry of the motion to be investigated and of the
end conditions it is known that
r = r 6 = r_j_ = r t = r 5 = r 1 and r 2 = /  4 (141)
By utilizing these conditions, we obtain the first three equations for
(137):
5/i  4r 2 + r 3 = Ar x
4f! + 7r 2  4r 3 = Ar 2 (142)
2r 2  8r 2 + 6r 3 = Ar 3
where
\ 4
mffi
The remaining two equations of (137) are duplicates of the first two. The
lowest eigenvalue of (142) is found to be
I = 0.0718
and the corresponding eigenvector is (1, 1.732, 2). Hence
0L = 6 x 0.0718^ = 3.108
(4.11) VIBRATION OF ELASTIC BODIES 315
The true answers are known to be
fiL = 77 = 3.1416
TT 7T TT
r x : r 2 : r 3 = sin — : sin  : sin — = 1:1 .732 : 2
6 3 2
This represents an error of about 1 per cent in {$ and 2 per cent in oj, since
oo is proportional to /? 2 . The values for r at the pivot points happen to be
exact.
It is interesting to note that this method of approximation results in a
frequency value lower than the true value, whereas most other approxi
mating methods give a value higher than the true value. This is because
the two sides are not equal in the approximation (136), and
/?%" = (r i+2  4r, +1  6r,  4/Vi + r t _ 2 ) + e
The error e is approximately 28
In a problem in which r is of the opposite sign to r" the error term makes
the finite difference approximation to r"" too small. Physically, it means
that the restoring force based on the approximation is too small or the
system is too soft. The result is a frequency lower than the actual.
(b) finite difference approximation for integral equation
In Art. 4.5 it was shown that the eigenfunction r(x) satisfies the integral
equation
— 2 r(x) = I G(x, !)m(!)r(£) d£
00* Jo
If we choose a set of equally spaced pivot points and integrate the right
hand side by either the trapezoid rule or Simpson's rule, we obtain a set
of algebraic equations in rvalues at the pivot points. Let us take again
the vibration of a uniform bar with hinged ends and divide the interval
into six segments. For a generic pivot point i the trapezoid rule yields
~AoJ 2 = h \2 Gi ° r ° + GilKl + ° i2r2 + Gl * rz + G * 4 '' 4+ Gi * r * + 2 Gl * r *)
For the first mode we can utilize (141) to obtain
2(G n + G i5 ) ri + 2(G, 2 + G i4 )r 2 + G t3 r 3 (143)
hpAo/'
28 Salvadori and Baron, Numerical Methods in Engineering, PrenticeHall, New York,
1952, p. 67.
316
THEORY OF MECHANICAL VIBRATION
(4.11)
It can be shown by theorems in strength of materials that for a simply
supported beam divided into sixths the influence coefficients are given by
G„ =
/z 3
36EI
(6  i)j(\2i  i 2  f) if i>j
Since G,
G H , this formula is sufficient for computing all G's. Let
/z 3
36£/
506
G 12 = G {
G 22 = 128/7
54
33
766
162b
^13 = ^31 = ^35 = ^53 = 78 £
^23 = ^32 = ^34 = ^43 = 1386
C7 14 = G 41 = G 25 = G 52 = 626
24
42
1206
G15
51
346
Substitute the values for the influence coefficient into (143) and simplify
42r ± + 69r 2 + 39r 3 = /^
69r x + 124r 2 + 69r 3 = Xr 2
78^ + 138r 2 + 81r 3 = /r 3
where
A =
18£/
h* P Aco 2 (f}hf
This time, since we are looking for the highest / so we can use the matrix
iteration method, we find
42
69 39"
r l.ooi
[241.71
69
124 69
1.76
=
425.1
78
138 81
2.00
483.0
= 241.7 x
1.000'
1.759
1.998
Hence
or
m*
241.7
/ 18 V A
^ =6 (wj) = 3  135
The error in the natural frequency is thus very small.
(4.11) VIBRATION OF ELASTIC BODIES 317
(C) COMPARISON OF THE DIFFERENT APPROXIMATE METHODS
Although the examples we have chosen do not include many complexities
that appear in practical problems solved by numerical methods, they do
give an idea of the algebraic and arithmetic processes required. When
handled properly, all numerical methods presented are capable of giving
good answers. They differ, however, in convenience for different prob
lems. The following general observations can be made regarding the
relative merits of the two methods presented.
(i) Difference equations derived from the differential equation formu
lation of an eigenvalue problem have the advantage of being easy to set
up. This is especially true when the boundary conditions are simple and
when there are no abrupt changes in mass distribution and elastic properties
in the system. When the number of pivot points is increased for accuracy,
the number of unknowns in each equation does not increase, although there
will be more unknowns and more equations to solve. The use of non
uniformly spaced pivot points is possible, but it is usually not convenient.
The smallest eigenvalue of the set of difference equations derived from the
differential equation corresponds to the lowest natural frequency of the
system. This is an undesirable feature because it is necessary to compute
the inverse matrix in order to use the matrix iteration method described in
Chapter 3.
(ii) The setting up of difference equations approximating the integral
equation requires extensive preliminary computation of the influence
coefficients or the values of Green's function. On the other hand, this
could be an advantage: (1) when a model of the system is available the
coefficients can be determined experimentally; (2) the treatment of
boundary conditions causes few special problems. The integration opera
tion is a "smoothing" operation. It is particularly suited for taking care
of abrupt changes in the interior of the system. For instance, if there is a
concentrated mass M attached to the midpoint of the bar, its existence can
be taken care of quickly by adding the term
M
to the righthand side of (143). The difference equation (137) cannot be
modified so easily. The use of pivot points not evenly spaced requires very
little modification of the trapezoid rule. When there are abrupt changes
in cross section, the convenience with which the integral equation can be
approximated with nonuniformly spaced pivot points becomes a distinct
advantage. The matrix equation representing the integral equation is
318 THEORY OF MECHANICAL VIBRATION (4.11)
already in the inverse form. That is to say, the largest eigenvalue corre
sponds to the lowest natural frequency so that the matrix iteration method
previously discussed can be immediately applied to determine the lower
modes of vibration.
(iii) In comparing these two methods with the energy method described
in preceding articles, we can say that in general the energy method is
capable of much more accurate results if we choose as many generating
functions as there are pivot points. In the study of twodimensional or
threedimensional systems the number of pivot points required to set up
difference equations is often too large even for machine computation.
In that case, the energy method becomes the only feasible one.
Exercises
4.1. Set up the differential equations of motion and the boundary conditions
for the following vibratory systems:
(a) The longitudinal vibration of a uniform bar, whose two ends are connected
to a stationary frame by two identical springs which exert axial forces on the
bar proportional to the end displacements.
(b) The lateral vibration of a cantilever beam having uniform cross sections
and carrying a concentrated mass at its unsupported end.
(c) The swinging of a heavy flexible chain hung by one end from the ceiling.
The other end is free.
(d) The longitudinal vibration of a tapered round bar, whose end diameters
are d and d L . The end with the larger diameter d is built into a rigid wall.
4.2. The formulation by differential equation of each of the following vibration
problems consists of two boundary value problems with interrelated boundary
conditions. Write the differential equations and specify the boundary conditions.
(a) The lateral vibration of a flexible elastic string initially stretched to a
tension S. A coil spring of constant K = S/3L connects the point x = L/3
to a stationary point to one side of the string, L being the length of the
string.
(b) The lateral vibration of a uniform beam of length L simply supported at
x = and x = L/3.
(c) The lateral vibration of a round bar with a builtin end and a free end.
The bar has a hole which extends from the free end to a depth of L/2 and
has a diameter of 3J/4, where L and d are the length and the outside
diameter of the bar, respectively.
4.3. Obtain by separation of variables the differential equations and the
boundary conditions governing the eigenfunctions belonging to the systems
described in Exercise 4. 1 .
4.4. A uniform bar with a fixed end and a free end is set into longitudinal
vibration in the following manner. A constant axial force P initially exists at
the free end of the bar, which is stationary. At / = this force is suddenly
removed. Determine the coefficient Q in (25). (Note that a, = 0.)
VIBRATION OF ELASTIC BODIES 319
4.5. Verify the characteristic equations and the eigenfunctions, given in
Art. 4.3(b) for lateral vibrations of uniform bars having various end conditions.
4.6. The vibration of each of the prongs of a tuning fork is approximately
the same as that of a cantilever beam. Design a tuning fork for the note C
(512 cps), which is to be made of steel weighing 0.286 lb per cu in. and having
Young's modulus of 29.5 x 10 6 lb per sq in. The effective prong length is 6 in.,
and the prongs are to have a rectangular cross section. What is the frequency
of the next highest note the fork may put out?
4.7. A vibrating steel reed having a natural frequency of 60 cps is commonly
used for the calibration of a stroboscope. If the reed is 2 in. long, what is its
thickness? Assume that the steel used has the properties given in Exercise 4.6.
4.8. A simply supported uniform beam of length L carries a concentrated
mass at its midpoint.
(a) Show that the characteristic equation for symmetrical modes of vibration
is
^0(tan 6  tanh 6) = 2
where fi = ratio of the concentrated mass to the mass of the beam
f
(b) Find the lowest eigenvalue for // = 2.
(c) A given function f(x) is to be expanded into an infinite series made up
of the eigenfunctions of the system. Show how the coefficients of this
series are determined.
4.9.
(a) Show that Green's function for a stretched string with x(0) = and
x(L) = is
G(x, £) = (L ~JK for < x < £
and
G(x, £) = ~{L  x) for £ < x < L
(b) Formulate by integral equation the vibration of a string with a mass
distribution given by
/ 7TX\
P A(x) = w l 1  sin — I
m(x) =
4.10.
(a) Obtain Green's function for a uniform bar of length L, which has a builtin
end at x = and a free end at x = L.
(b) If the right end of the bar is connected to a stationary frame by a linear
spring* so that E[ r(L) _ Kr(L) =
Green's function, different from that in (a), must be used in (68). However,
the same Green's function can still be used if we modify (68) to read
1 [ L
— r{x) = G(x, £)r() d£ + f{x)
<o* Jo
What is the function fix)']
320 THEORY OF MECHANICAL VIBRATION
4.11. The static deflection curve of a cantilever beam under a uniformly
distributed load is
u (x) = O 2 + 6L 2  ALx)
in which C is a constant. The function u(x), which is an admissible function,
can be used to approximate the first mode of lateral vibration of the beam.
Compute Q((u)) according to (72) and compare it with a^ 2 determined by (35).
4.12. The frequency of the first mode of vibration of the beam in Exercise 4.8
can also be approximated by Rayleigh's quotient for the static deflection curve
due to a concentrated force at the middle. Show that by the use of (96) and this
deflection curve we can obtain
«<*»*• 48£/ '
P ALU/i + 17/35)
Compare this approximation with the answer to Exercise 4.8(6) for u = 2.
4.13. Use a threeterm Fourier sine series approximation to find the approxi
mate answer to Exercise 4.8(6).
4.14. It has been pointed out that the eigenfunctions for a uniform cantilever
beam, as given by (36), can be used as the generating functions in the Ravleigh
Ritz method for the system shown in Fig. 92. In this procedure it is necessary
to evaluate
'L
[r^x)] 2 dx
\:
(a) Evaluate the foregoing integral for / = 1, 2, and 3, by the numerical
method. Note that although these integrals can be evaluated analytically,
a numerical method appears to be less timeconsuming, especially since
tables have been prepared for values of r { {x) by Dana Young and R. P.
Edgar, University of Texas Publications No. 4913, 1949.
(b) Use r l9 r 2 , and r 3 of (36) as the generating set to find the approximate
lowest natural frequency of the system in Fig. 92, assuming that
K Y = and K 2 =
EI
4.15. Utilize the results from (a) to find the approximate amplitude of steady
state vibration of a uniform cantilever beam at x = L, when a sinusoidal concen
trated force
fit) = pALg cos co f t
is acting on the beam at x = L. Assume that 2co f = coj = the lowest natural
frequency.
4.16. Given a differential equation,
L(r) = Ir
and a set of boundary conditions. Green's function associated with this
boundaryvalue problem G(x, ) can be defined as a function satisfying the
equation
KG) =
VIBRATION OF ELASTIC BODIES 321
and the original boundary conditions when x =£  and
L(G) = oo when x = £
Furthermore,
L(G) fife = 1
Show that
(aj) = AG(x,
Jo
satisfies the original boundaryvalue problem.
4.17. If the variables in polar coordinates are denoted by s and 6, the Laplacian
operator V 2 is known to be
a 2 l a l a 2
V 2 = 1 1
ds 2 s ds s 2 dd 2
(a) Write the differential equation of motion for the vibration of a uniform
thin circular membrane.
{b) Let the membrane be held fixed at its outer rim, where s = s , and let the
vibration be symmetrical with respect to the center. Show that the
eigenvalue problem associated with such vibrations is described by zero
order Bessel's differential equation
<Pr \_dr_
ds 2 s ds
where
The solution is zeroorder Bessel's function
r(j) = J (Ps)
(c) Show through the selfadjoint property of this eigenvalue problem that
sJo(fos)Jo(PjS)ds =0 / #y
Jo
where & and fa satisify
Ufr ) =
(d) By separation of variables, derive the eigenvalue problem for vibrations
that are not symmetrical with respect to the center of the membrane.
4.18. Show that the eigenvalue problem
DV = /9 4 r
r(0) = r'"(0) = r(L) = and r "(0) =
has no solution. Can you reason out physically that there is no solution?
4.19. Use the wave propagation concept to find the solution for Exercise 4.4
for < / < L/c. How can you relate this function with the indicial response
function ?
322 THEORY OF MECHANICAL VIBRATION
4.20. A uniform bar of length 4a is fixed at x = and free at x = 4a. At
/ = 0, the initial conditions for a longitudinal vibration are given by
u(x, 0) = x < x < a
u(x, 0) = 2a — x a < x <2a
u(x, 0) = la < x < 4a
and
(fL
(a) Show by concept of wave propagation that at t = 2a/c the displacement
is given by
u = —2"
u = (x — 2a) a < x < 3a
u = (4a — x) 2>a < x < 4a
u = —\x < x < a
(b) How would you find dujdt at t = 2a\c
4.21. A semiinfinite beam has a hinged end at x = and extends to £ = I x .
Show that (133) is the solution for the lateral vibration of this beam under the
initial condition
v (x, 0) = V{x)
(S)
s ■
4.22. Find the approximate solution for the first mode of vibration of a beam
with fixed ends :
(a) By finite difference equations approximating the differential equation.
(b) By finite difference equations approximating the integral equations.
4.23. Furnish the missing steps in the derivation of (134).
4.24. The boundary conditions for a uniform thin plate simply supported
along its edge are
w = and V 2 h> =0 on C
Show that the eigenvalue problem for the vibration of this plate is the same as
that for the vibration of a uniform membrane of the same contour.
Hint: The boundary value problem associated with (14) may be written as
(V 2  /5 2 )[(V 2 + £ 2 )r] =
in which £ 4 = ph/ET.
APPENDIX
An Outline of Matrix
Algebra in Linear
Transformation of Vectors
1. Introduction
A matrix is an array of numbers taken together as a single mathematical
entity. The dictionary meaning of the word matrix is "something in which
other things are imbedded." In mathematics an array of numbers, which
we call a matrix, also fits this description. Take for instance a set of linear
simultaneous equations
3^ + 4x 2
2x 1 + lx 2
5x,
= 1
= 4
= 3
(1)
The two things that characterize this set of equations or distinguish it
from other sets of linear equations are
the coefficients of the unknowns
and the constant terms
Therefore, in these two arrays of numbers a set of linear simultaneous
equations is imbedded.
323
p
4
5]
1 3
_2 7
1
r *!
4
3
324
THEORY OF MECHANICAL VIBRATION
(A.2)
If we wish to emphasize the fact that the unknowns are called x ± , x 2 , and
x 3 and that the constant terms appear at the righthand side of the equa
tions, we may write symbolically
4
51
m
r n
3
x%
=
4
7
1
_x 3 _
3
(2)
And if we wish to represent a set of simultaneous equations in general,
we may write symbolically
A x = c (3)
in which the symbol A stands for a square array of numbers (coefficients),
x for a column of numbers (unknowns), and c for a column of numbers
(known constants).
Thus we have a shorthand notation of a set of linear simultaneous
equations. This notation, called the matrix notation, achieves an economy
of thought as well as an economy in writing.
Notations alone, of course, do not make an algebra, which also deals
with rules of operation. In (3) we have more or less indicated that there is
a relationship denoted by the equal sign " = " and an operation denoted
by the multiplication sign "•". The question now is what other kinds of
operations can be defined and what algebraic properties do they possess.
For instance, is there a matrix and an operation "— " such that the
expression
A • x — c =
is equivalent to the one above?
equivalent to (3) we may have
Or, is there a matrix A 1 such that
c = x
Matrix algebra deals with questions of this nature.
2. Vectors in nDimensional Space
Although, historically, matrix algebra evolved from the study of linear
equations, for our present exposition we shall use geometrical models to
illustrate matrix operations. Such models have the advantage of revealing
at once both the computational aspects and the algebraic properties of
matrix operations. Geometrical models with more than three dimensions
are, of course, difficult to visualize. Fortunately, methods in analytical
(A.3)
AN OUTLINE OF MATRIX ALGEBRA
325
geometry allow us to extrapolate, so that we shall not have to visualize
more than three dimensions.
It is well known that by a suitable choice of coordinate system we
can represent any point in space by an ordered set of numbers, which we
call the coordinates of the point. Also, by joining the origin of the co
ordinate system to a point in space, we define a free vector, and the set of
numbers representing the coordinates of the point become the components
of the vector. The three things, a point, a vector, and a set of numbers,
therefore, are images of one another. We can thus use one kind of symbol,
such as p, to denote any one of these things, or we can use a set of n
numbers arranged in a column to denote the same things.
P =
Pi
P2
.Pn.
Such a column of numbers is called a column matrix.
3. Equality among Column Matrices
Two vectors are the same if and only if all their respective components
are equal. Two points coincide if and only if their respective coordinates
are equal. Hence we can define the equality between two column matrices
as follows. Let
P =
Pi
P2
and
The equality relation
implies
or, in general
J>nA \An\
p = q
Pi = qi P2 = a 2 m "Pn = q n
Pt= a i / = 1 , 2, • • • , «
An equality relation among column matrices is like the equality among
numbers in that it is reflexive, symmetric, and transitive.
326
THEORY OF MECHANICAL VIBRATION
(AA)
4. Addition of Column Matrices
(a) algebraic rules
It is known in vector algebra that if p and q are two vectors there is an
addition operation (parallelogram law) which yields another vector r. Or.
symbolically, p + q = r (4)
Two column matrices can also be added together to yield a third column
matrix. Furthermore, vectorial addition is known to be commutative and
associative. That is,
p + q = q + p
(p + q) + r = p + (q + r)
(5)
(6)
(B) ARITHMETIC RULES — PARALLEL COORDINATES
We have shown that two column matrices can be added together and
what the algebraic properties of the addition operation are. Now we may
ask, "How is the sum to be determined?" The actual numbers that make
up a column matrix, so that it represents a given vector, evidently depend
on the coordinate system chosen. The same vector is represented by
different columns of numbers in different coordinate systems. The question
of how to obtain from given p and q the column of numbers making up
r in (4) must also depend upon the choice of coordinate system.
The arithmetic rules of matrix operations are based upon the use of a
parallel coordinate system, in which the coordinate axes are all straight
lines. The scaling of each axis is uniform, but it can vary from axis to axis.
A threedimensional parallel coordinate system is illustrated in Fig. 100.
The vector p shown is represented by the column matrix
3"
4
2
A rectangular or Cartesian coordinate system is a special form of parallel
coordinate system in which the coordinate axes are mutually orthogonal
and the scaling is the same on all axes.
Vectorial addition in a parallel coordinate system is particularly simple.
It can be shown that the parallelogram law of addition is equivalent to the
following rule. If
Pi H\
p* q.
and
Pn
,?J
(A.5)
then
AN OUTLINE OF MATRIX ALGEBRA
p+ q
'(pi + ?i)
(P2 + ?2>
327
(7)
ipn + q n )_
In other words, each component of the vectorial sum is the sum of the
respective components of the vectors making up the sum.
Figure 100
Let us remark that although one can deduce the algebraic property of
addition, (5) and (6), from the arithmetic rule (7) the former has an
independent existence of its own.
5. Multiplication by a Scalar and Linear Dependence
If
we can write
P+ P
2p = q or p = iq
Thus a column matrix can be multiplied by a scalar, and the equality
q = ?P (8)
implies that in a parallel coordinate system
q i = 'kp i for all i = 1 , 2, • • • , n
328 THEORY OF MECHANICAL VIBRATION (A. 6 J
Geometrically, (8) implies that q and p have the same direction but differ
in length.
A vector r is said to be a linear combination of p and q if
r = ap + /5q
in which a and ($ are scalar quantities. Geometrically this means that the
line segments representing the vectors p, q, and r are coplanar. The
vectors p, q, and r are in the meantime said to be linearly dependent. In
general, a set of vectors p (1) , p (2) , . . . , p {i) is said to be linearly dependent
if for a certain set of scalar quantities, a l5 a 2 , . . . , a i? which are not all
zero, the following equation holds:
o^pW + <x 2 p< 2 >, • • • , <x,p<*> = (9)
The proof of the following lemmas is left to the reader as exercises.
Lemma 1. Any one of a set of linearly dependent vectors can be ex
pressed as a linear combination of the others.
Lemma 2. If p, q, and r are linearly dependent and r, s, and t are
linearly dependent, then p, q, s, and t are linearly dependent.
6. Linear Transformation of Vectors — Algebraic Rules
A set of n functions of n variables
Vl ~ r\\ x ii x 2> ' ' ' J x n)
2/2 == ^ 2\ X \'> X 2> ' ' ' 1 X n)
Vn — Fn\ X l> X 2> ' •> X )
can be considered as a functional relationship between two vectors
y = f(x) (io)
since, given a column matrix x, one can find a corresponding column
matrix y, and y may be considered as a vectorial function of a vector x.
There are many physical and mathematical models for the relationship
(10). We cite three such models for illustration.
(i) x is the location of a generic material point in a deformable body
before deformation, and y is the location of the same material point after
the body has deformed. In this case (10) describes how the body is
deformed.
(ii) x is the location of a generic spatial point in a steadystate fluid flow.
y is the velocity of the fluid particle occupying the point x. In this case
(10) describes the velocity distribution of the flow.
(A.6) AN OUTLINE OF MATRIX ALGEBRA 329
(iii) x is a column of numbers representing the coordinates of a fixed
point, according to a certain coordinate system, y is another column of
numbers representing the coordinates of the same point, according to
another coordinate system. In this case (10) describes a transformation of
coordinates.
In comparing (i) with (iii), we see that in one case the coordinate system
is fixed, whereas the point moves, and in the other case the point is fixed,
whereas the coordinate system changes. From a mathematical point of
view the difference in the situations hardly matters. We can, therefore,
always speak of (10) as representing a transformation of x into y.
The simplest example of a transformation is that represented by (8),
in which the vector q is obtained by "lengthening" p without a change in
direction. Or we can consider (8) as representing a uniform change of
scale factor of the coordinate system. Let us now conceive another class
of transformations, which is rather simple and yet of great utility in
analyzing physical and mathematical problem. These transformations are
called linear and are denoted by bold sans serif letters such as A, B, L, and
M. 1 A linear transformation is supposed to have the following properties:
(I) Every point in space has a transformation and the origin remains
fixed. Or, A(x), B(x), etc., are uniquely defined for every x and A(0) =
0, B(0) = 0, etc.
(II) Points lying on a straight line remain colinear.
Without going into the method of carrying out this transformation, we
can deduce the following lemmas successively from the properties postu
lated.
Lemma 1. Coplanar points remain coplanar because, unless a plane
remains a plane, some straight lines on the plane will become curved.
Lemma 2. Two intersecting lines remain intersecting, and two coplanar
parallel lines remain coplanar parallel. This follows if we take (i) to mean
that oo remains at oo and no finite point goes to infinity.
Lemma 3. A parallelogram remains a parallelogram.
Lemma 4. Linear transformation is distributive; that is
A(p + q) = A(p) + A(q) (11)
Proof
Let p + q = r
1 For the moment these symbols have not yet been tied in with the similar symbol in
(3).
330 THEORY OF MECHANICAL VIBRATION (A. 6)
Then, before transformation, p and q are two adjacent sides of a parallelo
gram and r is the diagonal in between. After transformation, the paral
lelogram is still a parallelogram, with A(p) and A(q) forming its two
adjacent sides and A(r) forming the diagonal in between. Hence
A(p) + A(q) = A(r) = A(p + q)
Lemma 5. Linear transformation preserves all linear dependence
relationships.
First we see that
A(2p) = A(p + p) = A(p) + A(p) = 2A(p)
and, by similar reasoning,
A(Ap) = AA(p) (12)
Hence, if
r = ap + /5q
A(r) = aA(p) + /?A(q) (13)
Lemma 6. A linear transformation followed by a linear transformation
is still a linear transformation. This is obvious from the postulated
properties of linear transformation. Symbolically, if
A(p) = q and B(q) = r
r is related to p by a linear transformation
r = C(p) = B[A(p)]
or
C = (BA) (14)
This is equivalent to a multiplication operation. We say by (14) that C is
obtained by premultiplying A by B.
Lemma 7. It follows from the foregoing that a succession of linear
transformations is still a linear transformation. Symbolically, if C = B A
and E = D C, then
E = [D(BA)] (15)
Furthermore, the multiplication operation is associative or
[D(B A)] = [(D B)A]
because, by the definition of multiplication operation,
[D(B A)](p) = D[(B A)(p)] = D[B{A(p)}]
[(D B)A](p) = [D B] {A(p)} = D[B{A(p)}]
Thus we can write (15) simply as
E = D B A
(A.6)
AN OUTLINE OF MATRIX ALGEBRA
331
Lemma 8. The sum of the results from two separate linear trans
formations on the same vector is also a linear transformation. This is to
say, if
A(p)+B(p) = u (16)
there is a linear transformation C, such that
C(p) = u
In other words, the relation between p and u, as defined by (16), is also a
linear transformation. To show this, we observe first that (16) satisfies
the property (I). All is left now is to show that (16) also satisfies property
(II). This can be done as follows:
The necessary and sufficient condition for three points p, q, and r to lie
on a straight line is that (see Fig. 101) there is a scalar parameter a so that
ap + (1 — a)q = r
Since A and B are linear transformations, according to (13),
aA(p) + (1  a)A(q) = A(r)
aB(p) + (1  a)B(q) = B(r)
Now if
A(p)+ B(p) = u
A(q) + B(q) = v
A(r) + B(r) = w
then by adding the equations in (17) and utilizing the foregoing equations,
we have
au + (1 — a)v = w
(17)
332
THEORY OF MECHANICAL VIBRATION
(A.7;
Hence the transformation (16) preserves the colinearity and is a linear
transformation representable by a symbol C. This completes the proof of
this lemma.
If we define
(A + B)p = A(p) + B(p) (18)
we may write
(A + B) = C (19)
Thus there is an addition operation among linear transformations.
Furthermore, from (5), (6) and (8) the addition operation is commutative
and associative.
A + B = B + A
A + (B + C) = (A + B) + C
Note that the operation (14) on the other hand is not commutative.
That is, in general,
A B(p) ^ B A(p)
or (21)
AB^BA
7. Linear Transformation in Parallel Coordinate Systems
In Art. 6 we made no mention of the method by which linear transforma
tion can be carried out. The algebraic rules were deduced strictly from
the geometrical properties of the transformation. To determine how a
linear transformation can be carried out, we refer again to a parallel co
ordinate system.
Let
and
2/2
Mr
where the elements in the column matrices represent the coordinates of x
and y in a parallel coordinate system. If x and y are related by a linear
transformation,
y = A(x)
then y l3 y 2 , . .'. . ,y n are linear combinations of x v x 2 ,
Vi = ^li^i + #12^2' ■ * , o, ln x n
Vi = fl 21^1 "T~ ^22^2' ' a 2n X n
«„.
(22)
(23)
(A.7)
AN OUTLINE OF MATRIX ALGEBRA
333
It is not possible to show here that a linear transformation in Euclidean
geometry must take the form of (23). However, one can easily verify that
(23) represents a transformation that satisfies all the requirements of being
a linear transformation.
The linear transformation (22) therefore embodies a set of linear
simultaneous equations relating the coordinates before to those after
transformation. These equations are imbedded in the matrix equation
y = A(x) = A • x = A x
Or
y*
a n a 12 • • • a
a 9 ,a
21" 2 2
a„,a.
• a
The rule of premultiplying a column matrix by a square matrix is given by
(23), or n
Vi = 2 a a x j / = 1, 2, • • • , w (23a)
3=1
where a {j is the element in the ith row and yth column of the square
matrix A. We can now deduce the following rules:
(i) If, for all x,
we write
A(x) = B(x)
A = B
This equality implies that the transformations A and B yield identical
results,
2 a a x i = 2 b a x o i = \,2, " ,n
j=i j=\
for all x t . In order for this to be true, we must have
i = 1,2, • • • ,n
j = \,2,,n
In other words, two n x n square matrices are said to be equal if and
only if all the corresponding elements are equal,
(ii) If, for all x, according to Lemma 8,
A(x) + B(x) = (A + B)x = C(x)
we write
A+ B = C
334 THEORY OF MECHANICAL VIBRATION (A.8)
This implies
n n
2 (an + b ij )x j = J c ij x j i = 1 , 2, • • • , n
3=1 3=1
or
i = 1,2, •*,«
«w + 0« = c u (24)
/= 1,2, •••,«
In other words, the sum of two « x n square matrices is also an n x n
square matrix, whose elements are obtained by adding the corresponding
elements of the original two matrices,
(iii) If for all x, according to Lemma 6,
B[A(x)] = C(x)
B A = C
n n n
2 2 b ik a kj x j = J c {j xj i = 1, 2,
j=lk=l j=l
n
i= 1,2,
• • , n
2* ^ik a ki = C ij
■ = l
7=1,2,
•  , n
we write
This implies
or
1=1.2. ",n
(25)
In other words, when two n x n square matrices form a product, which is
also an n x n matrix, the element at the /th row and y'th column of the
product is determined by the entire /'th row of the first matrix and the
entire y'th column of the second matrix, according to (25).
8. Summation Convention and a Summary
We note the following in all foregoing equations that describe the
relationships among elements of matrices in matrix operations:
(i) All subscript indices that are to be summed from 1 to n appear
precisely twice in a term.
(ii) All subscript indices that are not summed but are to take on any
value from 1 to n appear precisely once in each term of the equation.
Since this is always the case, we can simplify our notation by deleting all
summation signs and omitting notations such as / = 1, 2, . . . , n.
The convention then adopted is as follows:
(i) If an index appears twice in a given term, a summation process is
understood to exist without the explicit use of a summation sign. The
index is called a dummy index.
(A.8)
AN OUTLINE OF MATRIX ALGEBRA
335
(ii) If an index appears only once in each term of an equation (except
the term zero), it is understood that this index may take on any value from
1 to n. The index is called a. free index.
An equation such as (25) can be written without ambiguity as
£>ik a ki = c a
in which k is a dummy and / andy are free.
With this convention, we can summarize the result of the previous
articles by the following table:
Matrix Relation*
Relation Among Elements
x = y
x i = Vi
x + y = z
Ax = y
x i + Vi = z i
toi = Vi
A x = y
A = B
A + B =C
AA = B
a ij x j == Vi
Gil + bij = Cij
Xa^ = bn
AB = C
A BC = D
a ijbjk — c ik
QijDjkCkm = Cjm
* We have omitted parenthesis and dots in matrix products. Greek letters represent
scalar quantities.
The operations among matrices shown have all the properties of
arithmetic operation among numbers, except that multiplications are not
commutative. Thus we can deduce relations as follows:
where
(A + B)(x + y)
(A + B)(A .+ B)
A 2 = A A
Ax+Bx + Ay+By
A2 + AB+BA+B 2
and
B
BB
Also, if the elements of the matrices are functions of a scalar variable such
as time t, we may perform differentiation and integration processes with
respect to t, since they are the limits of processes involving only additions
and scalar multiplications. Hence
dt
(A+B)
dt
(AB)
d a d „
dt dt
a*)
B + AB
dt
336
Note that
Thus
THEORY OF MECHANICAL VIBRATION
(A.9)
■*■/
9. Inverse Operation
The linear transformation
A x = y
is supposed to be unique only in the sense that for every x there is a
unique y. It is, however, not necessary that a unique x be transformed
into every y. In other words, although every point is transformed into
only one point, two different points may be transformed into the same
point. If however, a onetoone correspondence exists in the transformation,
then the transformation has an inverse.
Let
A x = y (26)
If we are given y, x is to be found by solving the set of simultaneous
equations (23). Now, it is known that (23) will have a unique solution if
and only if the determinant
a u a 12 • •  a ln
a<„a 99   • a<
A =
a nl a n2
• • • a.
= W J ^
Thus the condition for the existence of an inverse to A is that A ^ 0.
We say that A is nonsingular if it has an inverse. Let the inverse of A be
denoted by the symbol A 1 . The inverse transformation of (26) is then
A 1 y = x
Evidently, if A 1 exists, it must also be a linear transformation.
The square matrix
"l ••• 0"
1 •••
1 =
(27)
(A.9) AN OUTLINE OF MATRIX ALGEBRA 337
is called an identity or unity matrix because it represents the identity
transformation
I x = x
where every point remains fixed.
From (26) and (27) we see that
A A 1 y = Ax = y = Iy
A 1 A x = A 1 y = x = I x
or
A A 1 = A 1 A = I
a ik a kj == l ij == °ij
(28)
where d {j is called Kronecker's delta 2 defined by
d tj = 1 when i = j
d tj = when i ^ j
In ^dimensional space (28) represent n sets of n simultaneous equations.
A typical set is
a 21 a 12 ~ 1 + a 22 a 22 ~ 1 + • • • a 2n a n2 ~ x = 1
«31«12 _1 + «32«22 _1 + • ' ' #3 A2 1 =
a m a i2 1 + ^2^22 x + ' ' ' a nn a n2 X =
If elements a tj are known, the elements a _1 of the inverse can be found by
solving these simultaneous equations. According to the socalled Cramer's
rule, the result is
= cofactorof^inKl
N
where
N = A # o
Although (29) is a concise definition for the elements of the inverse
of a nonsingular square matrix, the formula is not used for numerical
computation, except when the matrix has very few elements, such as a
3x3 matrix. This is because there are other schemes of numerical
computation that are less laborious. For our purpose it is sufficient to
know that a nonsingular square matrix has an inverse whose elements are
uniquely determined.
2 There is no reason to change i into d in (28), except that Kronecker's delta is an
established symbol.
338 THEORY OF MECHANICAL VIBRATION (A. 10)
From the definition of inverse matrix and identity matrix we have the
following equations:
If
C = AB
A 1 C = A 1 A B = I B = B
B 1 A 1 C = B 1 B = I
B 1 A 1 C C 1 = I C 1 = C 1
or
B i A i = c i
In other words
(A B)i = Bi A" 1 (30)
Similarly,
(A B C) 1 = C 1 (A B) 1 = C 1 B 1 A 1 (31)
10. Scalar Products and Transposition
So far, the matrix products that have been defined are A x and A B.
Products such as x y and x A have yet no meaning. In this article we
define the meaning of some new products.
(a) scalar product in cartesian coordinates
In vector analysis two vectors can form a scalar product with certain
physical or geometrical significance.
Let the following notations be adopted :
x and y = lengths of two vectors represented by x and y
p(x, y) = scalar product of the two vectors
6 = the angle between the two vectors
The geometrical meaning of scalar product is then
p(x, y) = x y cos 6 (32)
If the elements of the matrices x and y stand for the components of the
vectors in a Cartesian coordinate system with scale factor equal to unity.
it is known that
p(x, y) = x lVl + x 2 y 2 + ,••', x n y n = x { y t
In matrix notation we write
p(x, y) = [x x x 2 x n ]
y n .
= xy
(33)
(A. 10) AN OUTLINE OF MATRIX ALGEBRA 339
The components of the first vector are now arranged as a row, instead
of a column, to conform to the pattern of elementpairing procedure
previously established in defining products A x and A B. In these two
cases we take a row of the first matrix and pair its elements with a column
of the second matrix. The sum of the products of the pairs becomes an
element of the product. The symbol x denotes a row matrix that is
obtained by making the elements in the column matrix x into a row.
Evidently,
J x y = y x
(b) scalar product in general parallel coordinates
If the elements of the matrices x and y are the components of two
vectors referring to a general parallel coordinate system other than a
rectangular Cartesian system, the matrix product x y or y x is no longer
the scalar product defined by (32). To obtain p(x, y) in a general parallel
coordinate system, we may first transform the given coordinate system
into a Cartesian coordinate system. It must be realized that the scalar
product p(x, y), which is a scalar quantity of definite geometrical meaning,
remains invariant under any transformation of coordinates. In other words,
it is a quantity of definite numerical value, no matter what kind of reference
system is used for computation.
Let x and y be two column matrices whose elements represent the
components of two vectors in some given parallel coordinate system. Let
A be a square matrix representing a transformation that transforms the
given coordinate system into a Cartesian coordinate system with unit
scale factor. We use the article "a" in the foregoing sentence, instead of
the article "the," because the transformation is not unique, for two
coordinate systems may both be Cartesian but with differently oriented
axes. The matrices u and v defined by
u = A x and v = A y (34)
thus represent the components of the same two vectors referred to a
Cartesian coordinate system. Because of the invariance of scalar product
pi*, y) = />(u, v) = U v
the symbol u stands for a row matrix obtained by rearranging the column
matrix u into a row. To conform to the established pattern 3 of matrix
multiplication, we may write
u = [u ± u 2  ■ • U n ] = [XjX 2  • x n ]
= x A
#12^22 ' * a n2
#i«#9« * * * a nr ,
That is, by pairing the rows of the first matrix with the columns of the second matrix.
340 THEORY OF MECHANICAL VIBRATION (A. 10)
In longhand this equation is the same as the first one of (34), in which the
symbol A stands for a matrix that is obtained by rearranging the elements
of the matrix A in such a way that the rows of A become the columns of A
and the columns of A become the rows of A without otherwise disturbing
the order of things. In other words, the elements in A and A are related by
an = <*ji or a a = a H
1 11U5
p(x, y) = p(u, v) = uv = xAAy
Let
AA= M
(35)
p(x, y) = x M y
(36)
(C) TRANSPOSITION
(i) If A = B,then
B = A or
(ii) If A B = C,
(S).
B A = C or
Similarly
(AB)
(Ax) =
x A
The operation of converting the columns of a matrix into a row, and
vice versa, is called a transposition. The result of the transposition is
called the transpose and is denoted by a bar over the letter representing
the matrix transposed. Thus A is called the transpose of A and x is
the transpose of x. The following simple formulas pertaining to trans
position can easily be proved.
B A (37)
(A B C) = C B A (38)
(D) METRIC MATRIX OF A PARALLEL COORDINATE SYSTEM
If x represents the length of the vector x, according to (32),
x 2 = p(x, x) = x M x (39)
The matrix M is consequently called the metric matrix of the coordinate
system. For a given parallel coordinate system M is uniquely determined,
even though the matrix A in (34) is not. To prove this statement, let
us assume that there is another matrix, say N, that will yield the length
of a vector through (39); then
x N x = x M x
or
n^xj = m^x^
(A. 10) AN OUTLINE OF MATRIX ALGEBRA 341
Since x is arbitrary, the two sides can be equal only if the coefficients are
identical:
"ll = ™11 («12 + «2l) = ( m 12 + W 2l) "22 = ™22> ' ' •
or, in general,
(*« + «n) = (w« + ?%)
N + N = M + M ()
But, according to (35) and (37), a metric matrix is always symmetrical in
the sense that M = M because
M = AA M = (AA) = AA=M
So must __
N = N
Thus (40) reduces to
N = M
(e) orthogonality relation between two vectors
Two vectors are said to be orthogonal to each other if their scalar product
is zero. Geometrically, we see that
P (x y) = x y cos 6 =
implies that 6 = ±7t/2. For two vectors x and y to be orthogonal in a
general parallel coordinate system the condition is
xMy=yMx=0
In a Cartesian coordinate system,
M = 1
the orthogonality relation is expressed simply as
xly = xy =
Index
Absorbers, dynamic vibration, 138146
Accelerometers, 76, 7778
Admissible functions, 275277, 282286
Amplitude, 5
complex, 6, 8, 14
Amplitude modulation, 21
Amplitude ratio, 102, 111
see also Modal vectors
Analog computer, 227
Analogs, electrical, 221227
Analogous systems, 221
Analyzer, wave, 29
Aperiodic motion, 13, 115
Argand's diagram, 7, 8, 16, 45, 158, 159
Baron, M. L., 315
Boundary conditions, 238, 240
imposed or geometrical, 241, 248, 285,
292
linear homogeneous, 241, 251, 264, 270
natural or dynamical, 241, 248, 286,
292
Boundaryvalue problem, definition of,
240
Bromwich integral, 51
Cauchy's principal value, 49
Centrifugal force, excitation by, 125, 149
Chain systems, 212
Characteristic equation, 114, 116, 176,
189
see also Frequency equation
Characteristic values, see Eigenvalues
Circular frequency, 5
Collatz, L., 209, 286
Column matrix, 325328
Columns, buckling of, 302
vibration of, 300302
Compensation of instrument response,
78
Complex amplitude, 6, 8, 14
Complex damping, see Complex stiffness
Complex frequency, 14
Complex number representation, 6, 24
28
of damped oscillations, 14, 27, 88
of harmonic oscillations, 6, 2426, 85
Complex stiffness, 86, 88
Compound pendulum, 56
Concentrated mass on elastic bars, treat
ment of, 265268, 269
Constraints, 9, 169, 170, 238
effect on natural frequencies, 292294
Continuous frequency spectrum, 304, 305
Coulomb damping, systems with, 6268
Couplings, inertia and elastic, 106
effect on natural frequencies, 136138
Cramer's rule, 337
Crandall, S. H., 89
Critical damping, 14
resistance of galvanometer, 69
Critical speeds of shafts, 123128, 146
155
of the second order, 155
Curreri, J. R., 135
D'Alembert's principle, 224
Damped oscillation, 1214, 115116
343
344
INDEX
Damper, linear or viscous, 12, 114, 188
Damping, 13
Coulomb, 62
linear or viscous, 12
structural or hysteresis, 8289
Damping constant, 13
Damping of accelerometers, 78
Damping of galvanometers, 68, 69, 7071
Damping of vehicle suspension, 81
Damping ratio, 16
D'Arsonval galvanometer, 68
see also Galvanometer
Datum node, 224
Decrement, logarithmic, 15
Degrees of freedom, definition of, 169
Den Hartog, J. P., 65, 140, 153
Diagonalization of symmetrical matrices,
183
Difference equation, see Finite difference
approximations
Dirac function, 34, 267
Dirchlet's condition, 28
Dissipation, energy, 31, 188, 228
Divergence theorem, 270
Draper, C. S., 73
Duhamel's integral, see Superposition
principle
Dummy index, 173
Dynamical boundary conditions, see
Boundary conditions, natural
Dynamic state, 11
Dynamic vibration absorber, 138146
Edgar, R. P., 320
Eigenfunctions, 252
infinite series expansion by, 254, 265
267
orthogonality of, 260274
Rayleigh's quotient of, 277
Eigenvalues, 179
Determination by matrix iteration,
197207
maximumminimum characterization
of, 278279
minimum characterization of, 185, 278
negative, 302
see also Eigenvalue problem
Eigenvalue problem, of a differential
equation, 249260
of an integral equation, 271274
Eigenvalue problem, of a matrix equa
tion, 179
of a symmetrical matrix, 209
Eigenvectors, 179
see also Modal vectors
Elastically supported ends, 264, 269
Elastic body, configuration of, 237238
Elastic constants, 106, 171, 172, 176
Elastic coupling, 106, 137
Elastic energy, see Potential energy ex
pressions
Elastic matrix, 175, 208
Electrical analogy, 221227
Electromechanical systems, 227231
Enclosure theorem, 208209
End conditions, 241, 243, 247, 257258,
308
see also Boundary conditions
End mass, treatment of, 265267, 269
Energy dissipation, 30, 65, 83, 157
Energy methods, see Rayleigh's method
and RayleighRitz method
Energy relation, 10, 31, 186, 228, 229,
297
Equalization, of instrument response,
78
Error of finite difference approximation,
315
of galvanometer response, 72
Essential boundary conditions, see
Boundary conditions, imposed
Euler critical load, 302
Excitation, 40, 47, 48, 116
see also Signal
Expansions, infinite series, 238, 254, 265
267, 275, 288, 297
Finite difference approximations, 312—
318
for differential equations, 312315
for integral equations, 315316
Forced vibrations, of elastic bodies, 296
303
of severaldegreefreedom systems,
185188, 189192
of singledegreefreedom systems, 17
19, 23, 2829, 3139, 62, 6568, 83
86
of systems with Coulomb damping,
6568
INDEX
345
Forced vibrations, of systems with struc
tural damping, 8386
of twodegreefreedom systems, 116—
120
Foss, K. A., 192
Fourier series, 28, 29, 238, 255, 260
Fourier transform, 48, 309, 311*
Freedom, degrees of, 169
Free free vibration, 195
Free index, 173, 335
Free vibrations, of elastic bodies, 236
260
of severaldegreefreedom systems,
175178, 188189
of singledegreefreedom systems, 317
of systems with Coulomb damping,
6265
of systems with structural damping,
8689
of twodegreefreedom systems, 100
116
Frequency, 5, 10
beat, 22
circular, 5
complex, 14, 115
damped, 14
imaginary, 302
natural, 5, 102, 107, 112, 176, 292
Frequency equation, of bars in lateral
vibrations, 256, 257, 258
of bars in longitudinal vibration, 253
of chain systems, 213
of infinitedegreefreedom systems, ap
proximate, 281
of nonrotating shaft, 148
of rectangular membrane, 259
of rotating shafts, 151, 152
of severaldegreefreedom systems, 176
of twodegree freedom systems, 102,
106, 111
Frequency modulation, 21
Frequency ratio, 19
Frequency response, 23, 4143
of galvanometer, 7071
relation with indicial response, 4651
see also Steadystate response and
Transfer function
Frequency spectrum, continuous, 304,
305
discrete, 304
Friction, dry, 62
Functional, 275
Gain, 42
Galvanometer, equation of operation,
69, 230
frequency response of, 70, 71
optimum damping of, 71, 73
Generalized coordinates, 169173, 238
Generalized forces, 185186, 300
Generating functions, set of, 280
Graeffe's method, 116
Gravity effect on critical speed, 153155
Gravity pendulums, 5556
Green's function, 272, 273, 317, 319, 320
Green's theorem, 270
Gyroscopic effect on critical speed, 146
153
"Half" degree of freedom, 122
Hallowell, F. C, 89
Harmonic analyzer, 29
Holonomic systems, 170
Holzer's method, 212220
Hysteresis damping, see Damping, struc
tural
Hysteresis whirling, 156160
Identity matrix, 203, 337
Imaginary frequency, 302
Imaginary pivot points, 313
Imposed boundary conditions, 241, 248,
285, 292
Impulse response, defined, 36
of an elastic body, 303
of severaldegreefreedom systems,
187, 191, 192
of singledegreefreedom systems, 36,
37, 38
relationship with frequency response,
48, 49, 50, 51
Impulse response matrix, 191, 192
Incident wave, 308
Indicial admittance, see Indicial response
Indicial response, defined, 32
of an elastic body, 303
of severaldegreefreedom systems,
191, 192
of singledegreefreedom systems, 36,
37, 38
346
INDEX
Indicial response, relationship with fre
quency response, 48, 49, 50, 51
Indicial response matrix, 191, 192
Inertia constants, 106, 172
Inertia coupling, 106, 107, 137
Inertia matrix, 175
Infinite elastic body, 304311
Infinite degrees of freedom, 238, 281
Infinite series formulation, 287290
Influence coefficients, 108110, 208, 271,
316
Initial conditions, 5, 12, 13, 14, 28, 32,
35, 103, 115, 116, 178, 189, 191, 240,
254, 306, 311
Input, see Signal
Instruments, seismic, see Seismic instru
ments
Integrable constraints, 169
Integral equation formulation, 271274
Interior pivot points, 313
Inverse Laplace transform, 51
Inverse Nyquist locus, 46
Inversion of matrix, 336338
Iteration, matrix, 197202
Kinetic energy, 10, 106, 170, 284
Kinetic energy expressions, for elastic
body, 284, 287, 289, 290
for several degreefreedom systems,
171, 172
for singledegreefreedom systems, 11
for twodegreefreedom systems, 106
Kimball, A. I., 83, 89, 156
Kirchhoff's law, 221, 223, 224, 230
Kronecker's delta, 287, 337
Lagrange's equation, 106
Lagrangian description of deformation,
237
Laplace transform, 44, 51
Laplacian operator, 270
Lateral vibration of bars, 245248, 255
258, 309311
Lazan, B. J., 83
Lees, S., 73
Leonhard, A., 50
Lin, C. C, 116
Linearization of systems in small oscilla
tions, 5558, 171173
Linear systems, definition, 31, 171
Linear transformation of vectors, 179,
328, 334
Locii, transfer, 4546
Logarithmic decrement, 15
Logarithmic spiral, 16
Longitudinal vibrations of bars, 240243,
252255, 305309
Longitudinal waves, 305309
Loop current, 221, 222, 224
Lovell, D. E., 83, 89, 156
Lowest natural frequency, determination
of, 200202
MacDuff, J. N M 135
Magnification factor, 19, 23, 42
Matrix iteration, 197202
Matrix notation, 174
Maximumminimum characterization of
eigenvalues, 278279
Maxwell's reciprocal theorem, 110, 147,
272
Membrane, vibrations of, 248, 258260
Method of finite difference, see Finite
difference approximations
Metric matrix, 340
Minimum characterization of eigen
values, 278
Modal matrix, 178
Modal vectors, 177
Modes of vibration, see Principal modes
Mohr's circle, 107
Moving coil instruments, 68
Moving external force, 299
Myldestad, N. O., 89
Natural frequency of vibrations, of elas
tic bars and beams, 253, 256, 257, 258
of light beams and shafts carrying
masses and disks, lateral, 111, 134,
147149
of severaldegreefreedom systems, 175
of shafts, torsional, 212220
of singledegreefreedom systems, 4
of stretched membrane, rectangular,
260
of twodegreefreedom systems, 107
Natural modes, see Principal modes
Negative spring, 58
Network analogy to vibration systems,
221227
INDEX
347
Neumark, S., 89
Nonholonomic systems, example of, 170
Nonintegrable constraints, 170
Nonlinear systems, approximate analy
sis, 5562
Nonsingular matrix, 336
Normal coordinates, 181
Normalized modal vectors, 177, 210
Norton, A. E., 160
Nyquist diagram, 45
Nyquist locus, 46
Oil whip, 160161
Optimum damping of accelerometers, 78
of dynamic absorbers, 145146
of galvanometers, 7073
Optimum design of dynamic absorbers,
141146
Orthogonality of eigenfunctions, 260272
of modal vectors, 108, 181182, 197, 210
Orthogonality relation, defined, 260, 261,
341
Orthogonalization, 183
Overall transfer locus, 46
Overdamped systems, 13
Parallel coordinate systems, 326
Partial differential equation of motion of
elastic bodies, 239
Particular integral, 39
Pendulum, compound, 56
simple, 55
springloaded, 56
torsional, 52
Period, 5
Periodic forces, 28
Phase difference, 6, 23
Phase plane, 11
Phaseshift distortion, 78
Phase trajectory, 11
Phase velocity, 11
Pian, T. H. H., 89
Pickups, vibration, 7478
Piecewiselinear systems, 5968
Pivot points, 312
Plates, thin, vibration of, 249, 271, 322
Positive definite systems, definition of,
192
Potential energy expression, 10
of beams carrying weights, 134
Potential energy expression, of elastic
bodies, 93, 287, 290
of singledegreefreedom systems, 10
of semidefinite systems, 192, 193
of several degreefreedom systems,
171, 172, 174
of twodegreefreedom system, 105,
106
Potential forces, 185
Principal coordinates, 104, 118, 180, 181
Principal modes, 102, 104, 175
Principle of superposition, see Superposi
tion principle
Propagation of waves, 305308
Quality factor, 15
Rayleigh's dissipation function, 188
Rayleigh's method, 10, 132135, 185
Rayleigh's quotient, defined, 111, 184,
275
as function of amplitude ratio, 111
of admissible functions, 275
of essentially admissible functions, 284
of infinite series, 276
of vectors, 111, 184
stationary values of, 112, 132, 185,
277278
RayleighRitz method, 279292
Reflexion of waves, 308
Reluctance pickups, 75
Repeated roots of frequency equation,
123, 196197, 260
Resonance, 19
Response time, 72
Robertson, D., 160
Rotating shaft, vibrations of, 123128,
146161
Rotating vector, 79
Runge, C, 23
Salvadori, M. G., 315
Scalar product of vectors, 338341
Scarborough, J. B., 29
Seismic instruments, 7379
Seismograph, 7374
Selfexcited vibrations, 82
Semidefinite system, 192196
Semiinfinite elastic body, 304311
Separation of variables, 249252, 259
348
INDEX
Signal, 40
Soholnikoff, I., 46
Space variable, 249, 250
Spectrum, frequency, 304305
Stability of elastic systems, 302
Stable equilibrium, 171, 192, 302
Standing waves, 308309
Static deflection, defined, 19
Stationary properties of eigenvalues,
112113, 184185, 277278
Stationary values of Rayleigh's quotient,
112, 184, 277278
Steadystate response, 18, 39
of singledegreefreedom systems, 17
24, 2829
of severaldegreefreedom systems,
189190
of twodegreefreedom systems, 117—
119
with Coulomb damping, 6568
with structural damping, 8385
Step function, unit, 32
Stieljes integral, 34
Stiffness numbers, see Elastic constants
String, vibration of, 244245
Successive reduction of eigenvalue prob
lems, 206207
Summation convention, 173, 334
Superposition principle, 3136
applied to elastic body, 303
applied to severaldegreefreedom sys
tems, 187188, 190191
Suspension of vehicle, 7981
Sutherland, R. L., 22
Symmetrization of general eigenvalue
problems, 209212
Tensor notation, 173
Timeinvariant systems, 33
Timephase angle, 5
Timevariable, separation of, 249252
Timoshenko, S., 140
Torsional vibrations, 52, 212220, 243
Transducers, seismic, see Seismic instru
ments
Transfer function, 43, 44
Transfer locus, 43, 46
Transmissibility, 42
Transient response, 18, 3941
see also Indicial response
Triangular matrix, 211
Unit impulse, 34
Unit step function, 32, 46
Unity matrix, 337
Utube, oscillation of liquid in, 5254
Vectorial addition rules, 326327
Vectorial diagram of steadystate solu
tion, 27
Vehicle suspension, 7981
Velocity of longitudinal waves, 306
Velocity pickups, 76, 77
Velocity sensitive, 76
Veubeke, B. M. F., de, 200
Vibration absorbers, 138146
Vibration controls, 135138
Vibrometer, 76
Wave equation, 248
Wave phenomena, 304311
Work done by damping forces, 30, 65,
83
Work done in terms of generalized co
ordinates, 186
Work done per cycle, 3031
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